Juan Pablo Angel looked to have secured three points for Aston Villa despite having two second-half penalties brilliantly saved by Edwin Van der Sar
But even after Andy Cole had blown the chance to score from the penalty spot for Fulham the game was not dead - and captain Clark grabbed a dramatic stoppage-time winner.
Villa made three changes to their starting line-up as Thomas Hitzlsperger, Ulises De La Cruz and Carlton Cole stepped in, but there was no place in the side for new signing Eric Djemba-Djemba, who began the next step in his career on the subs bench.
Fulham brought Zesh Rehman and Papa Bouba Diop back into their starting XI, and it was the home side who threatened to take the lead within 20 seconds.
Tomasz Radzinski picked the ball up on the right side of the Villa penalty area and hit a fierce shot, but visiting goalkeeper Thomas Sorensen comfortably smothered his effort.
The opening ten minutes of the game provided few chances, and David O'Leary's men first threatened when Lee Hendrie fired high and wide.
Hitzlsperger headed wide for Villa moments later, but the game was still moving from end to end as neither team could take control.
It was on 15 minutes that Fulham produced the best move of the first half. Diop played a ball towards the box which Cole dummied superbly for Luis Boa Morte.
The Portuguese winger played a quick one-two with Clark before poking a right-footed effort towards goal, but Sorensen produced a good one-handed stop.
The Aston Villa goalkeeper almost handed Fulham the lead on 32 minutes as he fumbled Moritz Volz's cross, but Radzinski was unable to convert the rebound from close range.
The rest of the half fizzled out without note - but the second period provided much more explosive action.
Within two minutes of the second half Villa should have taken the lead - only for Lee Hendrie to write himself into football's book of howlers.
Nolberto Solano hit a shot-cum-cross along the six-yard area, but somehow Hendrie managed to steer his side-footed shot over the crossbar. It seemed much easier to have scored.
Even the Villa midfielder looked shocked, but the visitors refused to lose heart and eventually they gained their reward.
Hitzlsperger swung in a corner and Angel stole in front of Zat Knight to head the ball home. Clark desperately tried to clear the ball off the line, but his efforts were in vain and the travelling Villa supporters celebrated behind the goal.
Fulham attempted to strike straight back, and within five minutes they had the perfect chance to do so.
Referee Chris Foy adjudged Olof Mellberg to have tripped Boa Morte in the penalty area and pointed straight to the penalty spot.
The Aston Villa players argued furiously, but to no avail, and Cole stepped forward. The striker was looking to net his 11th goal of the season, but he dragged his effort wide of Sorensen's right-hand post to the dismay of the home fans.
Cole and Radzinski were replaced by Brian McBride and Collins John as Fulham searched for a new dimension in their attacking options.
But it was Aston Villa who were presented with a chance to double their lead on 75 minutes as Foy pointed to the penalty spot for the second time within quarter of an hour.
Volz appeared unlucky as Villa sub Luke Moore went down in the area with the pair shoulder to shoulder. But Van der Sar produced a fantastic right-handed stop to deny Angel his second goal of the evening.
Djemba-Djemba finally appeared for his first showing in a claret and blue shirt with 12 minutes left, and soon Villa had yet another chance from the penalty spot.
Moore went down again, this time under the challenge of Liam Rosenior. But the superb Van der Sar saved to his right-hand side to deny Angel yet again as the evening's penalty curse continued.
Fulham refused to give up and deep into stoppage time their battling qualities paid off.
Boa Morte's shot rebounded back off Hitzlsperger and Clark hammered an unstoppable drive into the bottom corner of Sorensen's net to steal a point and leave Aston Villa devastated.