The victory moves Motherwell three points clear of Celtic in second place.
The winger headed in a Tom Hateley cross after eight minutes but the visitors should have had the victory wrapped up before enduring a nervy finish.
Well midfielder Keith Lasley headed off the post in the first half before Hibs goalkeeper Graham Stack made a brilliant double save to deny both Stephen Craigan and Michael Higdon after the break.
As promised, Well manager Stuart McCall gave a start to Steve Jennings despite the midfielder’s recent arrest over alleged irregular betting patterns.
Hibs boss Colin Calderwood deployed the in-form Garry O’Connor up front after the striker successfully fought against a two-match ban over suggestions he dived to win a penalty in the recent win over St Johnstone.
The home side struggled to get out of their own half during the opening exchanges with Motherwell’s slick passing a factor in the away side retaining possession for long spells.
It was no surprise when McCall’s team went ahead after eight minutes when Murphy ghosted into the box to find the net. Murphy found a space between defenders Victor Palsson and Sean O’Hanlon to nod Hateley’s deep cross past Stack from close range.
Hibs were struggling to get going and Stack did well to push a Steven Hammell shot wide for a corner before then saving Tim Clancy’s near-post header from Hateley’s corner.
The Leith side had a rare sight of goal when Junior Agogo’s shot fell to O’Connor following Craigan’s block but the marksman hit a wild effort wide.
Motherwell should have been two ahead after 23 minutes when Hammell produced a teasing cross that found the unmarked Lasley, but the midfielder’s downward header came back off the post.
Clearly unhappy at his team’s first-half display, Calderwood introduced Danny Galbraith and David Wotherspoon for Palsson and Martin Scott at the break.
Winger Galbraith’s energy in midfield seemed to make a difference and he played his part in a half-chance for Agogo. The former Manchester United trainee passed to Ivan Sproule and the Northern Ireland cap sent a low drive into the box but his cross was just out of Agogo’s reach.
Nicky Law then won the ball back off Galbraith in a move that ended with Stack saving with his feet to deny Higdon after the striker was set up in the box by Murphy.
Motherwell looked dangerous on the counter-attack as Hibs upped their bid for an equaliser. Higdon and Chris Humphrey exchanged passes before the towering striker just missed the target with a low drive from 18 yards.
Stack then kept his side in the game with an incredible double save. His one-handed block kept out Craigan’s six-yard header from Hammell’s free-kick before he then somehow managed to parry Higdon’s point-blank shot.
From Hammell’s corner, Higdon powered a header off the bar. For all their possession and chances, Motherwell knew their lead was still fragile.
Wotherspoon had a free-kick from 25 yards tipped over the bar by Darren Randolph. Galbraith then found himself in space in the box after being teed up by Agogo but he lashed a strike into the side-netting.
Motherwell were on the back foot as the minutes ticked by and were relieved to see Agogo drill a low drive past the far post.