Richardson was the driving force behind Sunderland's attacking play
Sunderland produced a superb display but could not find the goal they deserved as Fulham escaped with a point to end a run of three straight defeats.
Kieran Richardson and Djibril Cisse came close to giving Sunderland the lead, while Zoltan Gera wasted two first-half chances for the hosts.
One Richardson free-kick hit the woodwork three times and another was disallowed for a foul in the wall.
Cisse volleyed on to the bar before Fulham's late pressure came to nothing.
Roy Keane, the Sunderland manager, will wonder how his team left London with anything other than three points, while his Fulham counterpart Roy Hodgson is unlikely to be too displeased by the result but has much work to do with his team.
Fulham remain a place and two points above the relegation zone and Sunderland move up to 12th following defeats for Middlesbrough and Wigan.
The hosts' absence of confidence was apparent from the outset after their recent results as runs went unchecked and passes horribly astray.
Home goalkeeper Mark Schwarzer was tested inside two minutes when Cisse's shot from an acute angle in the right channel had to be palmed behind for a corner.
Sunderland looked to attack at every opportunity, a similar approach to that which brought the Wearsiders their first win at Craven Cottage for 56 years last season.
Two early mistakes by Fulham right-back John Pantsil gave Richardson the confidence to run at his opposite man.
Hodgson relieved to escape with point
After four minutes the former Manchester United player cut in from the left and unleashed a fizzing, low effort towards goal only for Schwarzer to expertly tip it to safety with his fingertips.
Fulham soon began to impose themselves with Danny Murphy and Jimmy Bullard working hard to orchestrate play from central midfield.
For a period their passing was slick and incisive and when Bobby Zamora delightfully flicked Bullard's pass into Gera's path the Hungarian ought to have given his side the lead.
Gera rejected an initial opportunity to shoot, rounded Sunderland goalkeeper Craig Gordon but allowed the ball to run wide and his eventual shot was cleared by Pascal Chimbonda.
The 29-year-old spurned another gilt-edged chance when he completely missed his kick from in front of goal after Andrew Johnson had crossed low from the left.
Backed by a large and boisterous travelling support, Sunderland continually looked to punish Fulham on the counter-attack and it almost paid dividends as Cisse stepped in from the left and rifled a shot across goal that shaved Schwarzer's left-hand post.
Fulham will have been delighted to take a clean sheet into half-time, especially after Richardson's first free-kick on 41 minutes.
The winger's beautifully curled shot hit Schwarzer's left-hand post, squirmed across the goal line, rebounded off his right and then his left again before eventually being cleared.
Sunderland started the second period as they finished the first, showing little inhibition and flowing forward in numbers.
Keane stays cool after disallowed goal
In the space of three minutes Dean Whitehead drove inches wide, Richardson saw a lob cleared off the line by Aaron Hughes and the same player found the net with a free-kick, only to see it ruled out because Chimbonda was backing into the Fulham wall.
In what was turning into a intriguing encounter, the momentum then swung in Fulham's direction and industrious work by Davies ended with Gordon repelling Bullard's stinging drive.
But clear-cut openings were few and far between for Hodgson's men thereafter.
Former Cottagers striker David Healy was introduced by Keane and his headed knock-down allowed Cisse to rattle the woodwork with a sweetly struck volley.
Andy Reid's low effort from the edge of the penalty area was turned round the post by a sprawling Schwarzer a minute later as Sunderland were frustrated once more.
Fulham manager Roy Hodgson on referee Keith Stroud's decision to disallow Kieran Richardson's free-kick:
"He had warned (Pascal) Chimbonda already for pushing in the wall and obviously kept his eye on the situation.
"Fair play to him because luckily for us he was able to focus on his job.
"We looked a bit jaded to start with but improved and the main thing was we
ended a run of defeats."
Sunderland boss Roy Keane: "We should have had a goal from Kieron Richardson's free-kick as it was clearly a goal but I do not want to criticise the referee too much. We just have to get on with it.
"Instead of talking about a cracking game we are going to be talking about a goal which was not allowed.
"We hit the woodwork three or four times but I am fairly pleased with a point because we could have been out there all night and probably not scored.
"In terms of chances created we deserved to win the game."
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