Matthew Morgan slotted two penalties in the first half for the visitors and Leigh Halfpenny recorded three in total for the Blues before a 69th-minute Dan Parks drop goal briefly gave them a 12-11 lead.
However it wasn't to be as Biggar was presented with two kicks right in front of the posts late on as the Ospreys maintained their pursuit of Leinster at the top of the RaboDirect PRO12.
The Ospreys stated their intent from almost the first whistle with teenage fly-half Morgan kicking two penalties before five minutes were even on the clock.
However that was to be it for the Ospreys in the first half as they failed to record a single point in the next 35 minutes with Cardiff fighting their way back into the game.
Morgan missed a third penalty attempt in the 19th minute and Blues wing Halfpenny slotted two in the space of five minutes to make it 6-6 at the break.
The Ospreys had the perfect chance to retake the lead at the start of the second half but Morgan missed a second successive penalty however they didn't have to wait long to take charge.
And it came in the form of a Bowe try in the 51st minute - the Irish international rounding off a sweeping Ospreys move and touching down in the corner.
Morgan couldn't convert and 11-6 soon became 11-9 then 12-11 to Cardiff as Halfpenny sunk a well-taken penalty and replacement Parks slotted a drop goal.
It was the first time the Blues had been in front but it didn't last long with fly-half Biggar firing the hosts back in front with an easy penalty with eight minutes left to play.
The Ospreys took control but had to wait until the last kick of the game to make the result safe - replacement Biggar adding the finishing touch with a penalty.
|Ospreys Score Card|
|Cardiff Blues Score Card|