Paul McManus looked to have earned the hosts their first point against Rangers at Gayfield Park since 1938 when he cancelled out Jon Daly's opener.
But Aird's late winner ensured Arbroath would end the day seven points adrift at the foot of League One following East Fife's 2-1 win at Forfar.
The champions survived an early scare when Paul Sheerin picked off a poor clearance from Arnold Peralta and teed up Kevin Nicoll, whose composure let him down as he fired over at the far post.
Bobby Linn dragged a low shot wide before Rangers carved out their first chance, with Nicky Clark heading wide from Peralta's cross, while the visitors had penalty appeals turned down by Euan Anderson after Alan Cook appeared to handle Lee McCulloch's drive.
Michael Travis nodded over from Sheerin's corner at the other end, and Anderson waved away home claims for a penalty when McManus went to ground under Sebastien Faure's challenge in the box.
Linn fired straight at Cammy Bell when well placed in the area before Rangers went ahead on 19 minutes when Daly rose to meet Aird's corner with a firm header past Sandy Wood.
Linn drilled into the side-netting as Arbroath mounted a spirited response, although they came within inches of falling further behind when Aird's rasping strike from 30 yards beat Wood but cannoned off the crossbar.
Wood then did well to beat away an Aird free-kick, and Rangers were again denied by the bar 30 seconds into the second half after a misdirected header by Travis from Peralta's cross.
But Arbroath were gifted an equaliser on the hour mark as Faure's header back to Bell fell well short and McManus nipped in to slip a neat finish under the visiting keeper.
Cook then headed over from Sheerin's centre as Arbroath threatened to inflict a first league defeat of the season on Rangers.
But Ally McCoist's side snatched yet another victory when Daly's mistimed shot drifted across goal for Aird to turn in from close range.
However, Arbroath should have snatched a point in the dying seconds when Nicoll flicked on Sheerin's free-kick for Alex Keddie, who somehow scooped over from a couple of yards when it looked easier to score.