And they did it in dramatic fashion, 16-15, with outside half Biggar proving to be the hero of the hour.
Lee Halfpenny got the scoreboard moving with an early penalty, but James Bird levelled matters on 20 minutes with the Blues' first point of the night.
Bird's kick signalled a wonderful 20 minutes for the Ospreys during which time Biggar and Dean Ronan bagged tries to give the home side a comfortable 13-3 interval lead.
However, it was all change in the second half as the Blues struck back with tries from tries from Rhys Dyer and James Thomas to lead 15-13.
That was when Biggar stepped up to the mark, dropping a goal from all of 40 metres to earn the Ospreys a crucial win.