Home captain Stuart Malcolm had gifted the visitors the lead when he carelessly handled and top scorer David Hamilton did the rest from the penalty spot.
But Burns made an even bigger gaffe three minutes from time when he sliced the ball high into his own net.
Alloa had not scored since late August but they started positively with defender Fraser Forrest meeting Ross Hamilton's corner at near post but heading just over.
Stranraer were second best in the opening 25 minutes but one long pass from Joe McAlpine suddenly turned defence into attack and Kenny Wright should have done better with the finish.
David Hamilton's 20-yard effort then tested home keeper Allan Creer and Stranraer finished the first half on top as McAlpine's sizzling drive was beaten away nervously by Creer.
Malcolm's mistake and David Hamilton's penalty opened the scoring shortly after the break and Creer made another important save as Michael Moore looked set to double Stranraer's lead.
Against all expectations Alloa dominated the last 20 minutes but the Wasps seemed fated not to equalise.
Graeme Brown clipped the bar with a header and was denied four times by keeper Scott Black.