Despite having much of the possession the Scarlets lost out in the try stakes 2-0 with Richard Fussell and Alun-Wyn Jones crossing in the second half for the visitors. All the Scarlets points came from the boot of Stephen Jones who kicked six of his eight penalties.
It was the Ospreys' third-straight win at the ground in a game watched by 13,707 spectators.
But with a majority of possession and greater territorial advantage the Scarlets deserved to have a bigger lead than the 9-6 advantage they held at the interval, especially as it took the Ospreys until the 28th minute to get into the opposition 22.
In an intense atmosphere the Scarlets launched a ferocious all-out attack on the Ospreys in the opening quarter .
After Jones missed a penalty shot in only the second minute he was on target three minutes later as the home side took a deserved lead.
Opposite number Biggar, who was out to prove a point in front of watching Wales coach Warren Gatland, levelled the scoreboard on eight minutes after wing George North had been adjudged to be in front of the kicker chasing a Tavis Knoyle up and under.
The only things missing from the Scarlets performance in the opening 40 minutes were a strong scrum and a try.
A Jones cross-kick found wing Sean Lamont, but he was slammed into touch by Lee Byrne, while Regan King broke through a well-marshalled Ospreys defence and Dominic Day nearly created a chance when he charged down a Biggar kick.
Jones put his side back into the lead with a well struck penalty close to halfway but the Ospreys were back level with a Biggar penalty six minutes later.
But either side of half-time Jones kicked two penalties to put the west Wales region into a 12-6 lead.
However, with the Ospreys first meaningful attack of the match they grabbed a converted try. Shane Williams took a quick tapped penalty up the left. The ball was moved right through Andrew Bishop, Paul James and Byrne with Fussell going in down the right, although Byrne's pass appeared to be forward.
But despite being stunned by the try the Scarlets went back in front three minutes later with a fifth Jones penalty.
Just before the hour mark the Ospreys honed out a second try with skipper Alun Wyn Jones crashing over in the corner after Williams and Tommy Bowe had got their side into the 22.
On 63 minutes James Hook entered the fray for his first rugby of the season after recovering from a shoulder operation.
There was still plenty of drama to the match as a sixth Jones penalty levelled the scored at 18-18 on the hour mark.
But five minutes from the end of time Biggar slotted over the vital drop-goal before Jones was just off target with a crucial 40-metre penalty attempt.
|Scarlets Score Card|
|Ospreys Score Card|