The result extends the Ospreys' winning streak over their local rivals to an impressive five games and puts Biggar and co just one place behind the Scarlets in league table.
"It was great to get the win", said Biggar, who kicked 11 points in Llanelli.
"The Scarlets have played some great stuff recently and played well again. But we found an edge in the second half which they perhaps haven't come up against until now.
"We're very happy but want to push on in all competitions."
Much had been made of the battle between Biggar and Wales rival Stephen Jones in the build-up to the game.
Both fly-halves put in accomplished performances but where Biggar landed a drop-goal in the 75th minute, Jones failed with a last-minute penalty attempt.
"Personally, I'm very happy with my form", continued Biggar.
"Obviously, it was a big test last week against Steve. It was a game of two halves. The Scarlets forwards were excellent in the first half and ours were in the second.
"I thought Steve played very well first half but then we managed to get a little bit of a stranglehold on the game after the break.
"No10 is a very easy position when you are on the front foot. But you can ask Steve, Jonny Wilkinson or anyone else - it's not a nice position to be in on the back foot.
"It was nice to score a few points. I think we controlled a half each. But we battled it out and that is what people want to see. Luckily we came out on top."
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