Biggar extended Ospreys' lead with a further two penalties in an eight-minute spell after the interval but Jimmy Gopperth kicked three penalties to draw Leinster level with 12 minutes to go.
At that point the leaders were favourite to take the win but Biggar regained the lead with a well-taken drop goal and then held his nerve to seal the victory with a penalty.
Leinster, now focused solely on the RaboDirect PRO12 following their elimination from the European Cup last weekend, took the lead on ten minutes.
Gopperth had already missed one penalty before he slotted over the first three points of the match.
Ospreys responded by going through the phases and after Biggar saw his long-range drop goal attempt come back off the post the Wales fly-half levelled the scores with a penalty.
Ospreys were straight back on the front foot and a break from the impressive Dan Baker earned a penalty when Leinster flanker Kevin McLaughlin was again penalised, but Biggar failed with his attempt.
Isaac Boss was penalised for obstructing Biggar as he chased a loose ball a moment later though and this time the No.10 was spot on to put Ospreys 6-3 up.
After 25 minutes Biggar attempted to increase his tally only to see a drop goal drift wide.
Ospreys extended their lead with a fine try on 33 minutes. First Biggar's long pass found Richard Fussell who in turn set Hassler free.
Leinster scrambled back to deny the winger but Ospreys patiently worked through the phases and after Hassler ran them deep into Leinster territory, Fussell showed quick hands from Biggar's pass to leave Hassler with a simple run-in.
Biggar converted to extend the lead to 13-3 but Leinster are not RaboDirect PRO12 leaders for nothing and powered their way back into the contest.
From the lineout Leinster got the maul going and worked their way over, Ireland prop Healy taking the plaudits and Gopperth converting to send his side in 13-10 down at half-time.
Biggar continued to keep the scoreboard ticking over with his first penalty of the second half coming after 43 minutes.
It was the same story seven minutes later when Shane Jennings was penalised for not rolling away with Biggar extending the lead to 19-10.
A Gopperth penalty gave Leinster their first points of the second half on 53 minutes and they should have crossed for a try five minutes later only for Dave Kearney to spill the Kiwi's wonderful long pass with the try-line gaping.
From the resulting lineout, Leinster regained possession and after Ospreys were penalised Gopperth reduced the gap to three points.
There was a sense of inevitability about Leinster's comeback and they duly drew level when Gopperth slotted his fourth penalty of the night with 12 minutes to go.
Yet Ospreys were always in the contest and after punching holes in the Leinster defence, Biggar stabbed over a close-range drop goal to re-take the lead at 22-19.
And the 24-year-old made the game safe with three minutes to go when he nailed a penalty from the right touchline to open up a six-point gap.
|Ospreys Score Card|
|Leinster Rugby Score Card|