Fulham, chasing a third consecutive away win, never really got control of the game at any stage and spent most of the time on the back foot.
But they defended brilliantly with Schwarzer a superb last line of defence. The keeper made one magnificent stop from Tuncay's header immediately after the break.
Boro were pouring forward at that stage and Schwarzer went on to make crucial saves from Stewart Downing and Tony McMahon to keep Fulham in the game.
Boro could hardly have done any more throughout the game to try to record the win they needed to keep in touch with the teams just above them.
They started well with Schwarzer making an early save to keep out a volley from Jeremie Aliadiere.
Fulham did threaten briefly around the half-hour mark and ironically produced the closest effort of the game when Andy Johnson cut inside David Wheater and hammered the ball against the foot of Boro's right-hand post.
After the restart it was virtually all Boro again with Fulham centre-back Brede Hangeland and Aaron Hughes making numerous clearances.
With Schwarzer making several vital saves, Fulham gradually took the sting out of the Boro attack and Brazilian record signing Afonso Alves was taken off in the closing stages, having failed to cause any problems whatsoever.
Boro had run themselves into the ground and, in stoppage time, Fulham began to realise that they might yet salvage the win they needed to bolster their own European ambitions.
But Bobby Zamora wasted one chance when shooting over the top from close range before Boro keeper Brad Jones made his best save of the game when diving at the feet of Clint Dempsey.