But with a resolute backline featuring Joseph Yobo and Sylvain Distin snuffing out any attacks Fulham may have been considering, Everton began to show more purpose going forward.
Jack Rodwell had the first meaningful chance when his shot from outside the box took a deflection off Aaron Hughes, but Fulham's debutant goalkeeper David Stockdale was up to the task.
Hodgson praises Fulham revival
Just after the half-hour mark, Cahill gave his side a deserved lead.
Danny Murphy was booked for a clumsy challenge on Steven Pienaar, and from the resultant free-kick wide on the left, Leighton Baines sent in a ball that Cahill - holding off the attentions of Etuhu - buried in the right-hand corner.
Only a superbly timed intervention from Hughes prevented Leon Osman extending the lead, as the Northern Irishman took the ball off the midfielder's toes when he was in a threatening position in front of goal.
With Bobby Zamora and Andy Johnson getting no change out of Yobo and Distin, most of the few threats Fulham had came from Clint Demspey, but his United States team-mate Tim Howard in the Everton goal was rarely troubled in a first half controlled by the visitors.
Fulham showed more enthusiasm and purpose in the first 10 minutes of the second half than they had in the whole of the first period, however, and it soon produced results.
Neville bundled over John Pantsil just outside the box, and when Murphy's freekick rebounded to Konchesky, the defender's low shot took a wicked deflection off Distin, leaving Howard with no chance.
Moyes reflects on poor second half
Everton lost composure when Neville was stretchered off with a worrying looking injury to his knee which turned out to be nothing worse than a gashed leg, and Fulham took advantage to go in front.
Johnson drew Yobo out of position on the edge of the box and laid the ball off to Duff, who cut in from the right wing to fire a shot low into the far corner.
Substitute Marouane Fellaini and Yobo both had chances blocked in quick succession as Everton went in search of an instant equaliser, but Stockdale did well to keep them out.
In a frantic finale, Everton keeper Howard was playing up near the half-way line and was almost caught out by a Dempsey lob, but Duff's fine strike proved to be the match-winner.
Fulham manager Roy Hodgson:
"We have beaten a good Everton team today.
"We showed more courage in the second half, it was less a war of attrition and we turned it around.
"Damien is a goalscorer and it's nice to have players like him and Clint Dempsey and Simon Davies who can support Andrew Johnson up front."
Everton manager David Moyes:
"The first half was quite tight, we might have shaded it a bit but credit to Fulham they did much better in the second half and we dropped.
"They were chasing a goal but we were poor, we allowed them the initiative.
"We couldn't do an awful lot about the deflection but what was disappointing was the free-kick that we gave away."
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