Considine was only on the pitch because right-back Joe Shaugnessy had been sent off for a reckless challenge on Zaine Francis-Angol but he headed home from Jonny Hayes' 83rd-minute corner.
Motherwell had an early John Sutton goal disallowed but they initially struggled to make any inroads against the 10 men, although they improved after the break.
Stuart Carswell missed a good chance and Shaun Hutchinson headed off the frame of the goal but Aberdeen otherwise defended superbly against a succession of cross balls.
The Dons, who moved into second place in the Scottish Premiership on Saturday, also lost Clark Robertson to injury to leave Peter Pawlett and Hayes as makeshift wing-backs but they showed immense resolve and the Irishman ran clear after a Motherwell corner in injury-time to seal a 2-0 win.
Motherwell brought back Henri Anier and Francis-Angol following their lacklustre defeat by St Johnstone and they had the ball in the net inside five minutes only for a belated offside flag to spoil their celebrations.
Sutton turned the ball home from six yards after a clearance had cannoned off Paul Lawson but linesman Graham Chambers eventually put up his flag after a delay of several seconds, although it looked the correct decision.
The other major first-half decision soon came as Shaughnessy slid in heavily on Francis-Angol, prompting an angry reaction from the home side and referee John Beaton quickly ran over before flashing the red card.
Ryan Jack dropped back from midfield and Aberdeen manager Derek McInnes made a change eight minutes later when he brought on centre-back Considine for centre-forward Calvin Zola and moved to a 3-5-1 formation.
Niall McGinn was soon scythed down by Hutchinson after Lawson's slip had put Motherwell in trouble but the defender escaped with a yellow card.
Motherwell were dominating possession inside the Dons half but they could not break through the two lines of white shirts and home goalkeeper Gunnar Nielsen, whose slip gifted St Johnstone their second goal on Sunday, survived a couple of scares before the break when he spilled separate crosses.
Aberdeen were temporarily reduced to nine men after Robertson went down in agony after an innocuous challenge by Hutchinson and he was soon stretchered down the tunnel as Scott Vernon came on.
Aberdeen had an escape three minutes after the break when Mark Reynolds appeared to bring down Anier as the Estonian ran out of the box but Beaton ignored the penalty appeals.
Aberdeen defended a series of corners and crosses and they survived the best chance of the game so far on the hour mark when Francis-Angol cut the ball across for Carswell but the unmarked midfielder could only shoot straight at Jamie Langfield from 14 yards and the goalkeeper blocked.
Motherwell continued the onslaught from out wide and Hutchinson headed off the junction of post and bar from substitute Lionel Ainsworth's 71st-minute corner.
But Aberdeen always carried a threat on the break and Pawlett forced Nielsen into a decent parry at his near post after some slack defending from the hosts.
And Motherwell were made to pay from the resulting corner when Considine rose at the back post to head home Hayes' delivery from close range.
Stephen McManus spooned a volley over as Motherwell pushed for an equaliser but McGinn twice came closer before Hayes netted with the last kick of the ball.