The Wales fly-half kicked five penalties, a drop goal and converted Jeff Hassler’s first-half try to see off the league leaders.
Ireland prop Cian Healy got Leinster’s try which was converted by Jimmy Gopperth, who also contributed four penalties.
Former Newcastle man Gopperth had already missed a straightforward penalty attempt when he got the visitors on the board in the 10th minute as they enjoyed the better of the opening exchanges.
But the hosts slowly but surely began to assert themselves and Biggar knocked over two penalties to give them the lead at the end of the first quarter.
Their try arrived just after the half-hour. Hassler had already been denied once by Rob Kearney’s covering tackle, but there was no stopping him when Richard Fussell – on his 100th appearance for the Ospreys – flicked on Biggar’s wide pass for him to cross in the corner.
Leinster’s response was rapid as the Irish province turned down two kickable penalties to go for the corner.
The Ospreys successfully defended the first surge, but the second rolling maul was more potent and Healy was at the bottom of the pile.
The home side started the second half strongly and they looked to have taken a decisive lead when Biggar added another two penalties.
But back came Leinster through Gopperth. The Kiwi knocked over a penalty and was frustrated to see wing Dave Kearney drop what would have been a scoring pass minutes later.
Two more Gopperth penalties got Leinster level, but Biggar took control with an opportunist drop goal and his final penalty sealing a crucial win.
