Biggar was pivotal in the tense but full-blooded encounter and kicked 11 points, including a late drop-goal, to secure the victory.
"It was great to get the win", said Biggar.
"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."
Biggar displaced Jones in the final Test of the summer tour to New Zealand but, despite his fine form, the 20-year-old has insisted that selection for Warren Gatland's autumn squad is only at the back of his mind.
"Of course Welsh hopes are in the back of everyone's mind", added Biggar.
"But as I said at the start of the season, my main aim is to be playing for the Ospreys.
"I am taking it one game at a time and anything else that comes would be brilliant.
"I would be hugely disappointed if I'm not in the Autumn squad but ecstatic if I was.
"But with the November games you are judged on what you do in Europe. If I can come through the next couple of weeks, I'd like to think I will be in contention.
"At the minute I'm really enjoying my rugby. There's another great challenge on Saturday and there's nowhere else I'd rather be."