Murphy swept home a spot-kick in the 87th minute to clinch a 2-1 victory for Liverpool after substitute Florent Sinema-Pongolle was brought down by Zat Knight.
It was harsh on Fulham, who had survived an early goal from Emile Heskey to level through Louis Saha.
But things got even worse for the hosts when Luis Boa Morte clattered into a nasty knee-high challenge on Pongolle in the 89th minute and got himself deservedly sent off.
Fulham started the game slowly and although Saha fluffed an early chance from a long Steed Malbranque ball, it was no surprise when Liverpool took the lead in the 16th minute.
Former Fulham favourite Steve Finnan curled in a fine left-footed cross from the right and Emile Heskey volleyed home an instant shot that gave keeper Mark Crossley no chance.
The goal seemed to wake Fulham up and the hosts, who were without the injured Edwin van der Sar and Alain Goma, came close to equalising from a Saha header.
Even the loss of Mark Pembridge because of injury - he was replaced by Martin Djetou - could not deter them and they equalised after 39 minutes when Moritz Volz thundered his way upfield.
The German defender began his run deep in his own half before allowing Malbranque to take over and cross low for Saha to side-foot home at the far post.
The same player almost made it 2-1 on the stroke of half time, but saw his fizzing low effort smothered by Jerzy Dudek.
The second half was evenly balanced, with Saha twice going close to grabbing the lead for Fulham.
But Liverpool carved out the better chances, with Vladimir Smicer only denied by a fine save from Crossley.
Liverpool boss Gerard Houllier took off Michael Owen in a bid to win the match, bringing on Pongolle.
The change did not go down well with Liverpool's travelling fans, but proved to be a master-stroke in the end as Pongolle clinched victory by winning a penalty.