Danny Coles put the visitors ahead early on and, after David Healy had equalised, Craig Hinton's towering header had looked set to take Rovers - and their 7,000 travelling fans - into the fourth round.
But Murphy had the final say with a 20-yard drive in a see-saw battle that underlines why Fulham are second-from-bottom in the top-flight.
After another lacklustre display, new boss Roy Hodgson faces a battle to restore morale to a side who appear destined for the Championship.
Hodgson made three changes to the side that lost to Chelsea on New Year's, Alexei Smertin, Hameur Bouazza and Healy coming in for Elliot Omozusi, Steven Davis and Diomansy Kamara.
Bristol Rovers boss Paul Trolloppe, who was part of two promotion winning sides during his five years as a player at Craven Cottage, handed on-loan Hull midfielder Danny Coles a start in place of suspended captain Stuart Campbell.
Rovers' fans outnumbered their hosts and it took under three minutes for them to start celebrating, as Coles reacted quickest to a loose ball in the box and, via a wicked deflection, found the bottom corner.
Clint Dempsey had a goal ruled out for a push while Paul Konchesky hammered a 30-yard free-kick narrowly wide, but they were rare positive moments for the hosts, who were struggling to break down the defence of Rovers, who are 19th in League One, 44 places below Fulham in the league pyramid.
But out of nowhere the Premier League side equalised. It came from a mistake by Aaron Lescott - older brother of Everton's England international Joleon - who sliced his clearance to Moritz Volz, allowing the German to put the ball on the head of the unmarked Healy, who finished comfortably for his first goal in 11 games.
The visitors were competing well, though, and after weathering a storm from the Premier League side they re-took the lead with 25 minutes left, Hinton climbing highest to head Chris Lines' free-kick past a static Antti Niemi.
Their lead lasted only nine minutes, though, as former Liverpool man Murphy was given far too much space and fired into the top corner from 20 yards.
Neither side was unable to find a winner, but Rovers deserved their draw and Trollope's men will look forward to the replay in ten days time with greater anticipation than the Premier League side.