In-form Albion clocked up their third win on the trot without conceding a goal to keep their slim promotion hopes alive.
Stirling named the same team for the fourth match running and stay third in the table, nine points behind second-placed Stranraer after beating the Honest Men for the first time this season.
The Binos started like a train and carved out four good chances in the opening minutes, with Paul di Giacomo, Robert Dunn (twice) and David O'Brien all going close.
The deadlock was broken on the half-hour mark when keeper Ludovic Roy could only push out Di Giacomo's angled drive from the left, and Dunn pounced to head his third goal in two games.
Two minutes later it was game over. Right-back Paul Hay floated a free-kick into the area from near the centre circle, and the ball deceived everyone to end up in the back of the net. Former Motherwell midfielder Kevin MacDonald claimed he got a touch to the ball, but Hay was standing firm after the match and he looked to have the better case.
Ayr looked a poor side as they slipped to their fourth defeat in the last five games. They improved after the break but by then it was too late, and O'Brien should have added a third goal for Stirling but was denied by Roy when clean through.