Midfielder Murphy was left virtually unchallenged to nod in a Jimmy Bullard free-kick of a largely dismal game at Fratton Park which left FA Cup Finalists Portsmouth nursing their fourth consecutive defeat before next week's Wembley final against Cardiff.
But Fulham did not care a jot for Pompey's plight. And boss Roy Hodgson was able to celebrate a remarkable escape having almost given up the ghost a month ago when they lost at home to fellow strugglers Sunderland.
Since then they have beaten Manchester City, Birmingham and now Portsmouth. Yet Harry Redknapp's side, even without the injured David James and Sol Campbell, had ample possession and opportunity to sink their visitors.
Jermain Defoe, Niko Kranjcar and Pedro Mendes all went close but they really did not threaten Fulham's brave defence too much.
Those Fulham fans who packed the Milton End of Fratton Park were down in the dumps long before half-time after hearing relegation rivals Reading and Birmingham had both gone ahead.
They knew they should have done the same but for a brilliant save by goalkeeper Jamie Ashdown, who was standing in for the injured David James for the third match in a row.
Fulham fans had to cheer former Reading man Ashdown, though, when Paul Stalteri's cross found Davies whose shot from eight yards was going just inside a post until Ashdown flung himself across goal to push it away.
Kranjcar then cut for a one-two exchange with Kanu before blazing over and then drilled a free kick through Fulham's wall to make Kasey Keller hang on tight.
Brede Hangeland's interception stopped Kanu's goal-bound shot and John Utaka and Pedro Mendes both shot wide.
Fulham leaned heavily on the creativity of Bullard and Davies' non-stop running while Brian McBride struggled to find a way past the Pompey defence even though Noe Pamarot deputised for the injured Sol Campbell.
Fulham's Clint Dempsey looked hopefully at referee Mark Clattenburg when he went down in Pompey box but Hermann Hreidarsson's challenge did not even appear to make contact and the official waved protests away.
Dempsey, to his credit, responded superbly when needed at the other end - stopping Defoe on the edge of the Fulham area with a marvellous tackle but the Pompey striker should have put his side in front before the interval he blasted over.
Defoe and Kranjcar both tested veteran Fulham keeper Keller with snap shots after the break and the American was lucky to be in the right place to keep out the latter's 30-yards piledriver in the 55th minute.
Stand-in skipper Sylvain Distin almost headed into his own goal during a wild melee and then Kamara's cross came off Pamarot and flashed just wide as Ashdown flung himself across goal more in hope than expectation.
And when Bullard's latest free kick sailed into the area in the 74th minute Pompey generously left Murphy with all the room and time he needed to steer a simple header home and send the fans behind the goal wild with delight.
An extra three minutes stoppage time at the end must have shredded a few nerves but Fulham but with Hangeland a rock at centre back alongside Irishman Aaron Hughes, never looked vulnerable.
