Bolo Zenden scored a late equaliser from the penalty spot to keep alive Middlesbrough's hopes of qualifying for next season's Uefa Cup.
The late goal was awarded after Edwin van der Sar appeared to catch Jimmy-Floyd Hasselbaink but just outside the Fulham penalty area.
Brian McBride had deservedly put Fulham ahead with a right-foot strike.
Luis Boa Morte earlier hit the crossbar with a free-kick, while Hasselbaink went close with a header.
The result was harsh on Fulham, who created most of the chances and played with real determination.
Fulham manager Chris Coleman had to be restrained at full-time after trying to confront FA Cup Final referee Rob Styles over his decision to award the late penalty.
Boro remain eighth in the Premiership, one point behind Spurs, but will need to show more goal threat if they are to snatch seventh.
Fulham dominated from the start and McBride almost gave them an early lead.
But the US international, having climbed superbly to meet Mark Pembridge's free-kick, could not quite keep his header on target.
Middlesbrough appealed for a penalty when Hasselbaink went down under pressure from Liam Rosenior but Styles rightly waved play-on.
And it was Fulham who continued to create the better chances against a team struggling to cope with a crippling injury list.
With Michael Reiziger, Stuart Parnaby and Tony McMahon all injured 38-year-old central defender Colin Cooper had been drafted in at right-back.
And with Chris Riggott sidelined with a knee injury, Ugo Ehiogu made his fifth appearance of the season in the centre of defence alongside Gareth Southgate.
Collins John almost profited from a poor clearance, the ball falling at his feet 15 yards from goal, but the striker blazed over.
And McBride, fortunate not to be flagged offside, had another chance but again failed to find the target when completely unmarked.
Fulham continued to show the greater penetration after the interval and Boa Morte clipped the crossbar with a curling free-kick.
Steve McClaren urged his team forward from the sidelines and Boro finally started to show some urgency.
George Boateng narrowly missed with a strike from the edge of the area, while Hasselbaink almost scored with a near-post header across the face of goal that drifted just wide.
Ehiogu finally forced Van der Sar into a save with a weak header.
But McBride appeared to have scored the winner with eight minute's left before the late penalty drama ensured a point for Boro.
Middlesbrough boss Steve McClaren:
"The players have done a great job throughout the season but it will be tough to finish seventh with the players we have left.
"Our football, especially in the first-half, was not good enough but the players kept going and never gave up.
"We improved after the break and really took the game to them."
Fulham manager Chris Coleman:
"It was not a penalty. If it was in the box Edwin van Sar would have picked the ball up. Why would he try to head it?
"The players are devastated. We did enough to win and should be going home with three points.
"It is a shame but there are a lot of positives to take and our luck has got to turn sooner or later."
Middlesbrough: Jones, Cooper (Kennedy 86), Southgate, Ehiogu, Queudrue, Parlour, Boateng, Zenden, Downing, Hasselbaink, Nemeth.
Subs Not Used: Doriva, Davies, Wheater, Knight.
Goals: Zenden 90 pen.
Fulham: Van der Sar, Rosenior, Knight, Goma, Volz, Pembridge, Radzinski, Clark, Boa Morte, John (Malbranque 82), McBride.
Subs Not Used: Bocanegra, Crossley, Jensen, Rehman.
Goals: McBride 82.
Ref: R Styles (Hampshire).