Peter Weatherson rescued a point for Morton four minutes into stoppage time after Ayr looked home and dry.
In-form striker Gareth Wardlaw fired Ayr ahead with his 11th goal of the season in a first half dominated by the home side.
He latched on to a flick from Jerome Vareille and fired home from close range to get Ayr off to a flyer.
Vareille could have doubled the lead but keeper David McGurn pulled off a stunning save just before the break to turn his 18-yard drive around the post.
Morton piled forward in the second half and substitute John Adam had a powerful drive superbly saved by Ayr keeper Mark McGowan.
Just when Ayr looked like holding on for their first win of the year, it all came unstuck in the 94th minute when Weatherson was left unmarked to slam home a high ball from David McGregor.
Furious Ayr boss Robert Connor hit out: "Of course I don't know where the referee got so much injury time from, but there's nothing much we can do about it."
Relieved Morton chief Jim McInally said: "Our boys showed a lot of heart to keep going until the end after we left ourselves a mountain to climb."