Ayr striker Jerome Vareille grabbed a hat-trick on his return from injury but still didn't finish a winner as Albion hit back to end a run of six straight league defeats.
Stirling remain bottom of the table, but this result will give them a bit more hope going into Friday's rearranged fixture with Alloa at Forthbank.
Albion suffered the worst possible start when Dougie Ramsey knocked the ball into the area and Vareille beat the offside trap to fire past on-loan Rangers keeper Robert Giacomi.
With eight minutes on the clock, Vareille struck again – squeezing a shot into the corner after a misplaced pass from Paul Hay landed at his feet.
Stirling got themselves back in it when Paddy Connolly fired home his fifth goal of the season after 13 minutes, and Hay atoned for his earlier error when he equalised with a stunning free-kick from 30 yards after 42 minutes.
Both defences looked equally fragile, and Ayr surged in front again five minutes after half-time when Rob Giacomi came for a cross he was never going to get and Vareille glanced in a header.
But Albion were not to be denied and they levelled 60 seconds later when Connolly set up Robert Dunn to lash home his first goal of the season.
Substitute Stewart Devine missed a great chance to give Albion their first win of the season towards the end, but he fired over with just keeper Mark McGeown to beat and a draw was about right.