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The two sides could not be separated in a closely-fought first half, with Biggar and Rory Kockott both kicking four penalties.
Ospreys looked to be edging ahead at the start of the second half as the fly-half continued his solid kicking performance by adding a further six points for the home side.
The visitors made a series of changes as they tried to get the breakthrough against a side that they beat last week, but Biggar was able to add a further three points inside the last 10 minutes to put the Welsh club more than a converted try ahead.
The result leaves Ospreys level on points with Northampton Saints in their pool.