Murray and Callum Davidson combined to help secure a St Johnstone win that means double trouble for Dunfermline's hopes of avoiding relegation.
A Liam Craig free-kick crashed against a Pars post before Murray swept home his side's 33rd-minute opener.
Callum broke clear to finish clinically two minutes later.
Paul Burns fired a reply in a rare attack after 73 minutes, but Lee Croft killed off the game with a fine third five minutes later.
And their first defeat in 18 years at at McDiarmid Park leaves the Pars three points adrift of Hibernian at the foot of the table after the Leith side's 3-1 win away to Kilmarnock.
Croft came into the Saints midfield in place of Chris Millar following the goalless draw with Aberdeen.
Austin McCann, Gary Mason and Martin Hardie were restored to the Pars' starting line-up in place of Paul Burns, Kyle Hutton and Kevin Rutkiewicz after their midweek defeat by Celtic.
Different personnel, the same result, with the Pars on the back foot from the start and again being pinned back in their own half, with the home side dominating possession.
Francisco Sandaza was looking lively and several dangerous crosses were fired in, but Dunfermline held firm.
St Johnstone were dealt a blow in 15 minutes when Jody Morris limped off, being replaced with Millar.
Dunfermline goalkeeper Chris Smith, who had been excellent at Celtic Park, pulled off a superb double save.
First, he dived to his right to keep out Lee Croft's drive and then somehow clawed Sandaza's rebound effort off the line.
St Johnstone kept pressing and Frazer Wright's goalbound header was blocked by Paddy Boyle.
Saints were inches away from taking the lead after 26 minutes, when Liam Craig's inswinging free-kick missed everyone and smashed off the post before being scrambled clear.
Sandaza's movement was exceptional and a lovely flick released Croft, but his low shot was gathered by Smith.
Murray Davidson was next to try his luck with a turn and low drive after 32 minutes, but Smith saved well again only to be beaten a minute later.
Croft's chip was headed into the box by Millar and Murray Davidson swept home after good work from Sandaza.
In St Johnstone's next attack, it was 2-0. Sandaza found space in the box and fed Callum Davidson with a sublime reverse pass.
The full-back raced clear on goal and made no mistake with his finish from eight yards.
Dunfermline manager Jim McIntyre had already replaced David Graham with Paul Burns before half-time and brought on Paul Willis for Pat Boyle during the break, then Andy Kirk was soon replacing Liam Buchanan as he looked to rescue something from the game.
Still St Johnstone dominated and Sandaza twisted and turned before shooting narrowly wide.
Lee Croft smashed a rising drive off the upright in 69 minutes as St Johnstone chased a third.
However, it was Dunfermline who grabbed an unexpected lifeline, when Burns' curling shot came through a sea of bodies and nestled in the corner.
St Johnstone regrouped and Craig sent one drive wide and had another saved by Smith.
But Saints would not be denied for long and Croft made it 3-1 with a precise finish from 12 yards after some great skill.
Cillian Sheridan then hit the post in injury time as the hosts ended a run of three successive home defeats and left the Pars with only one win since 5 November.
St Johnstone manager Steve Lomas: "If you look at budgets, we probably shouldn't be battling for top six. But we are and the players are a victim of their own success as there's an expectation now.
"But we aren't getting too far ahead at the moment. There are many more tests to come and we can't get ahead of ourselves. Football can change in the space of 10 seconds.
"It was an important win. We got a great point at Aberdeen last week and it was vital to build on that.
"It was a fantastic performance and the scoreline could have been even bigger than it was."
Dunfermline Athletic manager Jim McIntyre: "The overall scoreline flattered us. We were lucky to only be 2-0 down at the break.
"We have been praised for our fighting spirit in the last few months, but I didn't see any of that.
"We just weren't good enough. There are no excuses. It was our worst performance in months.
"We got what we deserved and, although it's been a heavy week, we are miles better than what we showed.
"Maybe the Celtic game in midweek took more out of them than I thought, but that's still not an excuse."