Biggar's 20 points against Ulster were enough to fire them to victory last week and the eight-cap Welsh fly-half was at it again at the Liberty Stadium.
The Irish province tried manfully to bag their first win against the Ospreys from seven attempts with 11 points from Biggar's opposite number Niall O'Connor and two scores from Gavin Duffy.
But the right boot of the Ospreys No. 10 was ultimately the difference between the two as the home side's 100 per cent start to the season continued.
Connacht dominated early proceedings but the territorial advantage was wasted and it was the home side who drew first blood when full-back Barry Davies crossed out wide and Biggar added the conversion.
Biggar had a chance to stretch the lead when the visitors gave away the first penalty of the night on 20 minutes but the Welsh international's effort came back off the upright and Connacht escaped.
The visitors burst into life when Duffy very nearly latched onto a Brian Tuohy chip through but the full-back didn't have long to wait to cross with a 24th-minute score in the corner.
O'Connor added the conversion to level proceedings but six minutes later the Ospreys were ahead when Biggar notched a penalty to make it 10-7.
And Biggar was also on hand to convert calmly when a Hanno Dirksen try on 33 minutes stretched the home side's lead further.
That margin was shaved when the Welsh region were penalised for entering a ruck from the side and O'Connor duly slotted the three points to make it 17-10 at the break.
The Connacht No. 10 had a golden opportunity to make it a four-point game immediately after the break but he failed with his penalty effort.
The visitors were handed a a great chance to reduce the deficit on 54 minutes when flanker Justin Tipuric was sent to the sin-bin.
They looked to have wasted it when they allowed Biggar to take his tally for the evening to ten points with another penalty but Duffy scored his second try of the evening just after the hour mark.
O'Connor missed the conversion to leave the score at 20-15 and thereafter the visitors peppered the Ospreys line but Biggar put the game beyond doubt with another penalty with four minutes remaining.
O'Connor's late drop goal looked like it might not be enough for a losing bonus point as Biggar took his personal tally to 16 with another penalty, but the former Ulster fly-half kept his nerve with the final kick of the game to ensure the Irish side would at least take one point from the encounter.
|Ospreys Score Card|
|Connacht Rugby Score Card|