Bosnia's defeat kept them third in European Group H with 11 points, two behind Greece who were held 0-0 at home to Estonia.
Cyprus, fourth, with 10 points, still have an outside chance themselves of qualifying for the first time.
Senad Lulic hit the upright early on and then Vedad Ibisevic wasted a good chance on the break in a bright start by the visitors.
Toni Sunjic put Bosnia ahead in the 33rd minute with an ungainly effort, the ball bouncing into the net off his knee at the far post following a corner by Miralem Pjanic.
There was much more finesse about their second 11 minutes later as Edin Visca squeezed the ball into the net after a delightful dinked pass from Sead Kolasinac.
Cyprus captain Dimitris Christofi pulled one back in the 63rd minute and there was more drama four minutes later when
French-born midfielder Vincent Laban beat Asmir Begovic with a powerful effort from the edge of the area to equalise.
Pieros Sotiriou completed the comeback in the 76th minute when he fired a volley straight at Begovic. The ball hit the keeper and flew into the net for a famous win for Cyprus.
(Writing by Brian Homewood,; Editing by Neville Dalton)
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