Scrum-half Rhys Webb also went over for the Ospreys, who were the last side to win at Munster in April 2010, which cancelled out an early Danny Barnes try while Ian Keatley failed to keep up with Biggar's accuracy.
After only three minutes Barnes brilliantly blocked Hanno Dirksen's attempted clearance, picked the ball up and went over the line following good work from Doug Howlett and Johne Murphy. Keatley sending over the conversion.
Biggar kicked his first three points on seven minutes after the Irish side were penalised from the scrum before Keatley put seven points between the teams again with a penalty of his own.
Biggar and Keatley then both hit the post with attempted penalties midway through the half as little separated the teams.
Despite Ospreys spending large parts of the first half inside Munster territory, the Irish side held firm until the final kick of the half when Biggar kicked his second penalty to reduce the deficit to four.
Biggar continued to punish Munster after the break slotting another three points four minutes in - prop Stephen Archer, who conceded the penalty, sent to the sin bin for repeat scrum offences.
Ospreys captain Justin Tipuric was also yellow carded for hands in the ruck but the woodwork again denied Keatley from the resulting penalty.
The Irish province's luck got even worse three minutes later when a Mick O'Driscoll pass in front of Ospreys' posts was intercepted by Barry Davies.
He ran almost the entire length of the field before feeding Richard Fussell who in turn gave it to the supporting Webb to go over. Biggar this time missed the conversion.
Just after the hour, Keatley fired over another three points to bring Munster within one before Biggar again hit the woodwork, this time from a drop goal, but he made no mistake with the last kick of the game to secure their unbeaten record and place at the summit of the table.
|Munster Rugby Score Card|
|Ospreys Score Card|