Motherwell moved 10 points clear of Bank of Scotland Premier League bottom side Dunfermline with their second away win in four days.
The Pars dominated the opening spell but Marc Fitzpatrick burst forward to give Well the lead in the 26th minute.
Well grew in stature as the hosts missed more chances and Scott McDonald fired a superb second 11 minutes from time before Jim Hamilton was sent off for elbowing former Fir Park team-mate Stephen Craigan.
Pars striker Mark Burchill returned from injury to replace the suspended Stevie Crawford and Freddie Daquin came in for Greg Ross.
Well defender Paul Quinn replaced Jim Paterson, who received a red card in the 2-1 victory at Kilmarnock along with substitute David Clarkson.
Graeme Smith blocked Daquin's close-range effort in the third minute following a sweeping move by the hosts and some hesitant defending.
The Pars should have taken the lead in the 12th minute when Burchill and Gary Mason both missed a dangerous cross by Daquin after the Frenchman had skipped past Quinn.
Moments later, Burchill hooked a volley on to the bar from 20 yards with Smith scrambling.
Well soon had another escape when Martyn Corrigan cleared Phil McGuire's header off the line after a Scott Muirhead corner.
Muirhead then cut in from the left and saw his low shot smothered by Smith and Well finally threatened midway through the half when Elliot lashed a 25-yard volley just over.
And Well took the lead against the run of play in the 26th minute when Fitzpatrick burst on to Quinn's through ball and slotted the ball past the onrushing Dorus De Vries.
Moments later, Darren Smith missed a great chance for the second when De Vries brilliantly pushed away his effort after Elliot had sent him clear.
Souleymane Bamba and Burchill both came close from the edge of the box as shell-shocked Dunfermline tried to respond, but Well defended in numbers for the rest of the half.
Owen Morrison replaced the injured Daquin as the first half ended.
Burchill missed a one-on-one moments after the break when he placed a shot over Smith but inches past the post.
Hamilton headed a yard wide from Darren Young's cross and Burchill's first touch let him down as a half-cleared cross fell to him 10 yards out.
The Pars were winning the midfield battle but Well looked dangerous when they attacked and McDonald just failed to get enough strength on his hooked shot to trouble De Vries from a Brian Kerr cross.
Muirhead missed a glorious chance when he failed to get a decent contact on his volley from eight yards after Scott Wilson had turned back Mason's high free-kick.
Richie Foran was handed a yellow card after catching Mason with an elbow as the pair challenged for a high ball.
Wilson was raging with the Irishman and, seconds later, lunged in on a wild tackle on Elliot, which sparked a melee and earned him the same punishment.
The Pars had lost their attacking edge and Motherwell were looking increasingly dangerous on the break.
McDonald fluffed a shooting chance from 10 yards after a slip by Greg Shields.
But the Australian made no mistake in the 79th minute when he ran at the home defence, cut inside and smashed the ball into the top corner.
It was all too much for Hamilton, who led with his elbow in a terrible challenge on Craigan and was sent off despite the Northern Irishman getting up quickly.
McDonald again cut inside Wilson but De Vries blocked his low shot as the Steelmen comfortably played out the closing stages.