Biggar kicked an early penalty and converted the only try of the game, before failing with five successive penalty efforts - four of which were in the second half.
Flanker Justin Tipuric, the region's second youngest ever captain, claimed an opportunist 16th minute try as the Ospreys took a 10-9 lead into half-time.
But, in their 50th competitive at the Dublin venue, Leinster tidied up the mistakes and threatened with ball in hand - albeit with the Ospreys able to soak up the pressure.
Both sides made wholesale changes from their last outings, a flood of Ireland internationals returning on the back of Leinster's bonus point win over Connacht.
Tom Prydie was added to the Ospreys bench just before kick-off, replacing James Hook who was withdrawn due to a back spasm. Only four players were retained in the starting line-up from last weekend's defeat at Cardiff.
One of them, Biggar, punished an early maul infringement with his opening penalty, but a fine catch from Shane Horgan got Leinster moving in the right direction and Sexton's opener soon followed.
Solid defending from the Ospreys stopped Devin Toner and Brian O'Driscoll short of the try-line, before Sexton kicked Leinster into an 11th minute lead on a soggy pitch.
It took a well-timed tackle from Luke Fitzgerald to thwart Rhys Webb after a nippy tap and go from the Ospreys scrum half, but the visitors touched down from the very next phase.
Eoin Reddan had his clearance kick blocked down near the Leinster line, the dropping ball bounced free from Ian Evans' grasp and Tipuric sniped in to score under the posts.
Biggar tapped over the conversion but watched in horror when the ball moved on the tee just as he connected with what looked a straightforward penalty attempt on the half hour. His scuffed kick bounced off the underside of the crossbar and Leinster cleared.
Leinster had the better of the first half's final ten minutes, with Reddan and Horgan sparking a counter attack from deep. The Ospreys frustrated them at the breakdown, until Duncan Jones went offside and Sexton closed the gap to a single point.
Biggar's luck deserted him again when he struck the post from a 42-metre penalty as the second half got underway. But Sexton rewarded his forwards for a good bout of play, clipping over his fourth penalty success.
The two kickers miscued when presented with their next chances, and then Webb's throwing away of the ball before a Leinster lineout allowed Sexton make it 15-10.
Biggar failed with two more attempts - from long range and then out wide on the right - and the Ospreys, despite a late rally, were left clutching their second losing bonus point in seven days.
Fitzgerald looked the most threatening of Leinster's runners, but spilled possession and slow ruck ball prevented Joe Schmidt's men from striking for a try.
|Leinster Rugby Score Card|
|Ospreys Score Card|