And after Welsh record tryscorer Shane Williams scored his second try in his final appearance for the region with three minutes remaining, Biggar coolly slotted the conversion from the touchline to earn a one-point win.
It was Biggar who had given the Ospreys an early lead with a penalty, cancelled out by Ireland fly-half Jonathan Sexton.
Biggar added another three points but Sean Cronin crashed over for the first try to put Leinster back in front.
After a third penalty from the Ospreys fly-half, Leinster extended their lead when Isa Nacewa scored the first of two tries straight from the restart.
The Ospreys fought back early in the second half with a try from centre Ashley Beck but two more penalties from Sexton extended the home side's lead.
Williams then raced in for his first score of the day with an hour gone, only for Nacewa to score his second, Sexton's conversion making it 30-21.
But after another Biggar penalty, Williams danced his way over on the right, giving Biggar the chance to win it.
And having missed just one kick all day, he struck it perfectly to give the Ospreys the lead with just three minutes remaining, and they held on to deny Leinster a European and domestic double.