A game which was slow to warm up came to life after 20 minutes when Steve Marlet found himself with only Peter Schmeichel to beat after being played in by Steed Malbranque.
Marlet's shot had the sting taken out of it by the keeper, but as the ball was about to cross the line a brilliant overhead clearance by Richard Dunne was quickly matched at the other end.
Anelka wriggled past two defenders and was then upended on the edge of the box by Zat Knight. Kevin Horlock's low free-kick beat the wall, but not the diving Taylor, who first parried and then somehow punched the ball in the air similar to the famous save from Gordon Banks against Pele.
Before that Sean Davis had volleyed inches wide after he had taken the ball chest high from Malbranque's pass.
Taylor had also been called into action when he smothered Anelka’s close-range effort after Darren Huckerby had robbed Knight in midfield.
At the start of the second half both Huckerby and Marlet went close but Anelka missed an absolute sitter when he somehow nodded wide from barely three yards after a low cross had been missed by two Fulham defenders.
It mattered little when he finally broke the deadlock to leave Fulham facing a potential relegation battle.