Hodgson admitted before the match that he was more concerned about Saturday's visit of Manchester United in the Premier League, but he will surely be delighted with this win.
Basle had won all five of their previous European home games this season by two goals or more and Fulham were yet to win on their travels in the group stage.
But the Cottagers, whose line-up featured six changes from the one that started at Burnley last weekend, were worthy victors.
They got off to a bright start in snowy conditions at St Jakob-Park and might have opened the scoring inside a minute when Basle failed to clear Riise's right-wing cross and Jonathan Greening came close to poking home.
Fulham's 4-4-1-1 formation denied Basle the chance to settle on the ball in attacking areas while their three-pronged central midfield of Murphy, Greening and Dickson Etuhu dominated possession and dictated the game's tempo.
Murphy saw a half-volley well-saved before Frei stepped inside Stephen Kelly but a poor touch enabled Mark Schwarzer to foil the Swiss striker.
That was Basle's only clear-cut chance in a one-sided first half and Fulham were rewarded for their superiority when Riise curled in a delightful cross from the right, which an unmarked Zamora met with a clinical finish at the back post.
Basle were furious that referee Stefan Johannesson waved away appeals for a free-kick in the build-up to the goal but there was no excuse for their woeful marking.
Zamora, who had already seen a goal disallowed after collecting Riise's chip and slotting into the bottom corner, was causing the home defence no shortage of problems and deservedly doubled his side's lead three minutes later.
The 28-year-old diverted Schwarzer's long kick forward to Riise and the Norwegian stood up an inviting cross which Zamora met with a towering header for his ninth goal of the season.
Murphy was at the heart of Fulham's play all evening
With his side needed only a draw to progress, Basle coach Thorsten Fink introduced striker Federico Almerares and the attack-minded Xherdan Shaqiri at the interval.
And, despite Zamora testing home goalkeeper Massimo Colomba with a deflected free-kick early in the second half, the changes initially had the desired effect.
Cheered on heartily by their boisterous supporters, Basle were suddenly enjoying the lion's share of possession and got themselves back into the match.
Smalling handled an Almerares cross and, after some confusion over whether Johannesson had pointed for a corner or a penalty, Frei lashed his spot-kick into the top left-hand corner.
With Fulham showing minimal attacking intent, it looked as though a full-scale comeback was inevitable.
But the goal merely proved a wake-up call for Fulham, who reasserted control over proceedings and then stretched their lead as a 15-pass move ended with Gera applying a first-time finish to Kelly's cutback.
Basle continued to fight for their lives and Frei drilled a shot against the bar before Streller angled Valentin Stocker's corner past Schwarzer.
But Fulham held out and could even have grabbed a fourth when an unmarked Dempsey headed Riise's cross straight at Colomba.
Fulham manager Roy Hodgson: "Bobby Zamora carried on from where he left off in the Premier League. I thought he was very unlucky to have a first goal ruled out for offside, and he was a real thorn in the flesh of the Basle defence.
"I took him off as a precaution, he had a slight problem with his back. But he's got to be delighted with his performance.
"Bobby is one of our more popular players, he has a chant of his own and the fans seem to appreciate and like him.
"The one criticism is he hasn't scored enough goals but nine goals is a pretty good return for any forward. We were pretty good value tonight for the goals we scored and we could have scored more."
90:00+3:00 The final whistle is blown by the referee.
90:00+2:02 Effort from just outside the box by Valentin Stocker goes over the bar.
89:07 Inswinging corner taken by Alexander Frei, Headed effort from inside the area by Benjamin Huggel misses to the right of the
88:34 The ball is crossed by Bjorn Helge Riise, Clinton Dempsey takes a shot. Massimo Colomba makes a save.
87:29 The ball is swung over by Samuel Inkoom, Chris Smalling makes a clearance.
86:36 Assist on the goal came from Valentin Stocker.
86:36 GOAL - Marco Streller:Basle 2 - 3 Fulham Marco Streller finds the net with a headed goal from inside the six-yard box. Basle 2-3 Fulham.
86:27 Corner taken by Valentin Stocker from the right by-line,
85:57 Xherdan Shaqiri takes a shot. Mark Schwarzer makes a save.
85:26 Unfair challenge on Alexander Frei by Danny Murphy results in a free kick. Free kick crossed right-footed by Alexander Frei
from right wing, Chris Smalling makes a clearance.
82:42 Frederico Almerares sends in a cross.
81:28 Alexander Frei takes a shot.
79:57 Inswinging corner taken by Alexander Frei, Frederico Almerares produces a header from inside the area that goes over the crossbar.
79:30 Substitution Damien Duff on for Bobby Zamora.
Correction - 78:05 Substitution Pascal Schurpf replaces Carlos Alberto Carlitos.
78:05 Substitution Alexander Frei leaves the field to be replaced by Pascal Schurpf.
77:44 Clinton Dempsey produces a left-footed shot from deep inside the penalty box which goes wide of the left-hand post.
76:15 Assist on the goal came from Stephen Kelly.
76:15 GOAL - Zoltan Gera:Basle 1 - 3 Fulham Zoltan Gera finds the net with a goal from inside the penalty box to the bottom right corner of the goal. Basle 1-3 Fulham.
76:03 Stephen Kelly crosses the ball,
74:31 The referee blows for offside. Direct free kick taken by David Abraham.
73:36 Unfair challenge on Samuel Inkoom by Bobby Zamora results in a free kick. Samuel Inkoom takes the direct free kick.
72:27 Stephen Kelly produces a cross, clearance made by Cagdas Atan.
70:53 The ball is swung over by Samuel Inkoom, Close range header by Frederico Almerares misses to the right of the goal.
69:12 Substitution Clinton Dempsey comes on in place of Jonathan Greening.
67:35 The ball is sent over by Carlos Alberto Carlitos.
65:12 Frederico Almerares produces a cross, Header from deep inside the penalty area by Valentin Stocker goes harmlessly over the
63:26 GOAL - Alexander Frei:Basle 1 - 2 Fulham Placed penalty scored by Alexander Frei. Basle 1-2 Fulham.
Correction - 62:51 Aaron Hughes is adjudged to have handled the ball.
62:51 Handball decision goes against Aaron Hughes.
61:36 Cagdas Atan takes a shot. Comfortable save by Mark Schwarzer.
61:17 Alexander Frei decides to take the corner short.
59:54 A cross is delivered by Valentin Stocker, Frederico Almerares has a headed effort at goal from deep inside the area missing
to the right of the target.
58:12 Alexander Frei takes a shot. Blocked by Dickson Etuhu.
57:39 Valentin Stocker produces a cross, save by Mark Schwarzer.
57:02 Inswinging corner taken left-footed by Valentin Stocker from the right by-line, Chris Smalling makes a clearance.
55:38 The ball is sent over by Valentin Stocker, Mark Schwarzer makes a comfortable save.
54:09 Xherdan Shaqiri takes a shot from just outside the penalty area which clears the crossbar.
53:53 Xherdan Shaqiri delivers the ball.
52:01 Inswinging corner taken from the left by-line by Alexander Frei, Aaron Hughes manages to make a clearance.
50:24 Direct free kick taken by Jonathan Greening.
50:24 Booking Jonathan Greening shown a yellow card.
50:20 Free kick awarded for a foul by Frederico Almerares on Jonathan Greening.
49:34 Corner taken by Bjorn Helge Riise from the right by-line, Frederico Almerares makes a clearance.
48:57 Centre by Jonathan Greening.
47:55 Valentin Stocker takes a shot. Chris Smalling gets a block in.
45:30 Free kick awarded for an unfair challenge on Bobby Zamora by Valentin Stocker. Direct strike on goal from the free kick comes
in from Bobby Zamora, Massimo Colomba makes a save. Jonathan Greening takes the inswinging corner, clearance made by Samuel
45:03 Substitution Adilson Cabral leaves the field to be replaced by Frederico Almerares.
45:01 The referee gets the second half underway.
45:00+1:08 The whistle is blown to end the first half.
45:00+0:28 Booking Alexander Frei is shown a yellow card.
44:30 Assist by Bjorn Helge Riise.
44:30 GOAL - Bobby Zamora:Basle 0 - 2 Fulham Bobby Zamora scores a headed goal from close range. Basle 0-2 Fulham.
44:15 Bjorn Helge Riise delivers the ball,
43:56 The referee blows for offside against Alexander Frei. Mark Schwarzer takes the direct free kick.
43:34 Zoltan Gera concedes a free kick for a foul on Alexander Frei. David Abraham takes the free kick.
43:18 Bobby Zamora gives away a free kick for an unfair challenge on Cagdas Atan. Massimo Colomba takes the free kick.
41:39 Assist on the goal came from Bjorn Helge Riise.
41:39 GOAL - Bobby Zamora:Basle 0 - 1 Fulham A goal is scored by Bobby Zamora from inside the six-yard box to the bottom left corner of the goal. Basle 0-1 Fulham.
41:01 The ball is swung over by Bjorn Helge Riise,
39:51 Free kick crossed right-footed by Jonathan Greening from left channel, Berang Safari makes a clearance.
39:51 Booking David Abraham receives a caution for unsporting behaviour.
39:35 Unfair challenge on Danny Murphy by David Abraham results in a free kick.
38:56 Bjorn Helge Riise produces a cross.
38:23 Alexander Frei challenges Danny Murphy unfairly and gives away a free kick. Aaron Hughes takes the direct free kick.
37:01 The ball is sent over by Samuel Inkoom.
35:05 The ball is sent over by Jonathan Greening, blocked by Berang Safari.
34:19 The assistant referee flags for offside against Bjorn Helge Riise. Massimo Colomba takes the free kick.
Correction - 34:19 The referee blows for offside. Massimo Colomba restarts play with the free kick.
Correction - 33:36 Foul by Alexander Frei on Jonathan Greening, free kick awarded. Bobby Zamora delivers the ball from the free kick right-footed
from left channel.
33:36 Jonathan Greening fouled by Alexander Frei, the ref awards a free kick. The free kick is delivered right-footed by Bobby Zamora
from left channel.
33:24 The ball is swung over by John Pantsil, clearance made by Cagdas Atan.
31:02 Unfair challenge on Carlos Alberto Carlitos by Jonathan Greening results in a free kick. The ball is delivered by Alexander
Frei, Headed effort from deep inside the penalty area by Cagdas Atan misses to the left of the goal.
29:54 Free kick taken by Massimo Colomba.
29:54 Booking Booking for Bobby Zamora.
29:42 Free kick awarded for a foul by Bobby Zamora on Berang Safari.
28:58 A cross is delivered by Samuel Inkoom, Shot from just outside the box by Alexander Frei goes over the bar.
28:23 Bobby Zamora gives away a free kick for an unfair challenge on David Abraham. Direct free kick taken by Massimo Colomba.
24:44 Unfair challenge on Bobby Zamora by Cagdas Atan results in a free kick. Bjorn Helge Riise delivers the ball from the free
kick right-footed from right wing, clearance by Marco Streller.
23:27 The assistant referee signals for offside against Zoltan Gera. Cagdas Atan takes the direct free kick.
19:51 Alexander Frei takes a inswinging corner from the left by-line to the near post, John Pantsil makes a clearance.
19:22 Alexander Frei takes a inswinging corner from the left by-line played to the near post, Chris Smalling manages to make a clearance.
19:05 Samuel Inkoom sends in a cross.
18:27 Centre by Bobby Zamora.
17:54 The ball is crossed by Berang Safari, save by Mark Schwarzer.
17:20 John Pantsil produces a cross, clearance by Cagdas Atan.
15:59 The ball is crossed by Bjorn Helge Riise, Bobby Zamora takes a shot.
15:23 Bobby Zamora is caught offside. David Abraham restarts play with the free kick.
11:15 Foul by Carlos Alberto Carlitos on Jonathan Greening, free kick awarded. Jonathan Greening crosses the ball in from the free
kick, clearance made by David Abraham.
10:14 The referee blows for offside. Massimo Colomba takes the free kick.
8:42 Berang Safari delivers the ball, Marco Streller produces a header from inside the area that goes over the bar.
8:23 Marco Streller delivers the ball, clearance by Aaron Hughes.
6:13 The ball is crossed by Bobby Zamora, Danny Murphy takes a shot. Save made by Massimo Colomba. Jonathan Greening takes the
inswinging corner, clearance made by David Abraham. Jonathan Greening takes a inswinging corner to the near post, Cagdas
Atan manages to make a clearance.
5:04 Corner taken right-footed by Alexander Frei, Mark Schwarzer makes a comfortable save.
4:19 Inswinging corner taken from the left by-line by Alexander Frei, Chris Smalling makes a clearance.
2:28 Berang Safari fouled by Zoltan Gera, the ref awards a free kick. Direct free kick taken by Cagdas Atan.
1:55 Alexander Frei takes a shot. Mark Schwarzer makes a comfortable save.
0:35 Bjorn Helge Riise delivers the ball.
0:00 The match begins.
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