The League One champions took the lead through Jon Daly's 25th goal of the season but looked set for a slip-up when French defender Sebastien Faure undercooked a header back to goalkeeper Cammy Bell just after the hour mark.
That let in Paul McManus to score as the Red Lichties primed themselves for their first home draw against the Ibrox giants since 1938.
But Aird's late goal handed Ally McCoist's side the victory to maintain their 100 per cent record outside of Glasgow.
Suspension and injury were behind three of McCoist's five changes to his side.
Midfield pair Nicky Law and Ian Black, out with back and foot knocks picked up against Brechin last weekend, dropped out, as did banned defender Bilel Mohsni.
Callum Gallagher and Dean Shiels were also dropped as Faure, Kyle Hutton, Robbie Crawford, Arnold Peralta and Nicky Clark were given the opportunity to stake a claim for a starting berth ahead of next week's Ramsdens Cup final.
McCoist's men started sloppily and stayed that way for the 90 minutes. They survived a third-minute scare when Peralta's woeful clearance fell straight to Paul Sheerin and the Arbroath player-manager picked out Kevin Nicoll at the back post but his shot flew over Bell's crossbar.
Bobby Linn then dragged a low effort wide after racing clear of the visitors' back four but Rangers created their first opening after 10 minutes when Clark headed wide from Peralta's cross.
The Ibrox men then saw demands for a penalty turned down by referee Euan Anderson after Alan Cook handled from Lee McCulloch's strike.
Rangers continued to look fragile at the back, allowing Michael Travis clear room from 10 yards to nod just over from Sheerin's corner.
Arbroath then had a decent penalty appeal of their own rejected after Faure appeared to barge into McManus from behind, while Linn should have done better than to shoot straight at Bell after 19 minutes as the ball broke to him in the box.
But within seconds, Rangers had the lead as Daly powered home a firm header from Aird's corner.
The Ibrox side still did not look secure as they again allowed Linn to take aim at Bell's goal before breathing a sigh of relief as he buried the ball into the side-netting after 25 minutes.
Yet they could have extended their lead moments later as Aird drilled the ball off Sandy Wood's crossbar with a 30-yard strike.
Nicoll was the first man in the book a minute before half-time after body-checking Lee Wallace, while Aird saw the resulting free-kick palmed away by Wood.
The hosts almost handed Rangers a second 30 seconds into the second half as Travis headed Peralta's cross onto his own crossbar.
But it was a gift from Faure which saw Arbroath level.
The defender was under no pressure as he tried to head a high ball back to Bell but did not spot McManus lurking on the edge of the box and the striker made him pay as he slotted under the approaching goalkeeper.
There was another close call with 12 minutes remaining as Cook got onto the end of Sheerin's centre before putting his header over.
But it was Rangers who snatched a victory they barely deserved when Daly mishit substitute Gallagher's cross before Aird turned the ball into the net from close range.
Yet Arbroath should have levelled with the last kick of the game. A Sheerin free-kick was headed on by Nicoll before Alex Keddie somehow turned the ball over the bar from a yard out for an atrocious howler.