Aird fired the visitors ahead following a defensive mistake and, after Robert Thomson had clipped home a deserved equaliser, Clark headed the winner to make it 15 away wins from as many matches for the Glasgow side.
However, the opener required a touch of good fortune when home defender Jonny Brown slipped as he tried to deal with Ian Black’s long punt forward, allowing Aird to race clear and coolly place the ball into the bottom right hand corner of the net on 23.
Despite dominating possession, Rangers failed to create much in the way of clear cut chances during the majority of the opening 45 minutes – other than the goal.
Brechin even enjoyed a brief period of pressure towards the end of the first half, although visiting keeper Cammy Bell was rarely troubled.
The closest City came to scoring before the break came via a long range shot from Darren Petrie which Bell comfortably punched clear for a corner.
And the hosts continued where they left off early in the second half, equalising with a well-taken effort on 50.
From Greg Cameron’s low cross, Thomson reacted quicker than any defender to turn the ball home from close range.
But slightly against the run of play substitute Clark headed Rangers back in front, less than two minutes after replacing Black who was carried off the pitch with a nasty looking ankle injury.
An unmarked Clark found space on the edge of the six-yard box to nod home Jon Daly’s superbly weighted chip in the 67th minute.
The goal certainly rejuvenated Rangers who controlled the remainder of the match, with Aird just failing to extend the margin 14 minutes from time, while Lee McCulloch headed inches over on 82.
The result means the Ibrox club have now won 28 of their 30 League One fixtures this campaign, to head the table by a massive 32 points.