Substitute Callum Lang looked to have put Shrimps on the road to just a fourth away win of the campaign courtesy of his 91st-minute effort but Ahearne-Grant levelled things up as it ended 1-1 at the Checkatrade Stadium.
Reds - who dropped a place to 12th in the table following the draw - wasted numerous chances prior to going behind.
Lang's strike looked to have won it for Morecambe before Charlton loanee Ahearne-Grant's late, late effort was enough to ensure Crawley preserved their unbeaten home record in 2018.
A point leaves Reds five points adrift of Coventry City - in the final play-off place as things stand - with ten games to go.
Reds made one change from the side that were beaten at Forest Green Rovers last time out. Injured Jordan Roberts made way for midfielder Dannie Bulman.
Just as in the Sussex Senior Cup semi-final win over Saltdean United on Tuesday, Harry Kewell went without a recognised striker as Mark Randall operated as a 'false nine'.
Crawley saw a lot of the ball in the opening exchanges and created the first real opening. Randall's clever pass slid Karlan Ahearne-Grant in, only for his fizzed cross to be hooked clear by Luke Conlan after 20 minutes.
Reds then threatened from a corner two minutes later but Morecambe managed to clear the loose ball.
Crawley were looking most threatening from wide areas. This time full-back Lewis Young's superb, fizzed cross was cleared by Sam Lavelle on 28 minutes.
Morecambe, who'd managed just three away wins all season coming to Crawley, came close late in the half. Shrimps skipper Micheal Rose's long-throw was flicked on to Adam McGurk but his header went wide.
Reds made a lively start inside the opening five minutes of the second half. Captain Jimmy Smith's fierce drive was parried away by Barry Roche, then Enzio Boldewijn's glancing header went just wide.
Frontman Panutche Camara was introduced from the bench 25 minutes from time - going straight up top - which seemed to spark life in to Crawley.
A flowing move saw Boldewijn pick out Smith, he found Young whose neat cut inside but saw his goalbound strike was blocked by Kevin Ellison.
Reds were once again denied by some superb last-ditch defending from Morecambe after 73 minutes. Boldewijn found Ahearne-Grant - who fired towards goal from 15 yards out - only for Lavelle to block his goalbound effort.
Crawley were creating chances but lacking cutting edge. Boldewijn teed up another team-mate - this time skipper Smith - but he managed to fire wide from less than ten yards out.
Reds were then fortunate not to go behind with four minutes left. Midfielder Andrew Fleming was allowed too much time and his dipping 25-yard strike came back out off the crossbar.
Shrimps looked to have snatched all three points a minute in to second half stoppage-time. Substitute Callum Lang went clean through, chipping over the onrushing Glenn Morris to fire Morecmabe in front.
Reds finally found a way through to snatch a point three minutes later. Bulman picked out Ahearne-Grant and he slotted home from close range as it finished 1-1.
CRAWLEY TOWN: Morris; Young, Connolly, Yorweth, Doherty; Bulman; Boldewijn, Smith, Payne, Randall, Ahearne-Grant. Subs: Mersin, McNerney (Payne, 86), Djalo, Tajbakhsh, Lelan, Sanoh, Camara (Randall, 62).