Stevenson put the home side in front before the break with a firm header from Michael Wardrobe's cross.
But second-bottom Forfar pulled themselves level against the run of play when Andy Ryan finished off a swift counter attack.
However, Stevenson's overhead kick earned the Honest Men just a second win in six Scottish League One games.
Ryan Stevenson's brace gave Ayr United victory over Forfar Athletic and strengthen their grip on third place.