Neil Doak's men trailed 13-6 at the interval after a lacklustre first half, but roared out of the traps in the second period to score three tries through Ruan Pienaar, Luke Marshall and Craig Gilroy.
Pienaar also hit two penalties and two conversions, but Kristian Phillips did add a late score for the visitors to claim a valuable losing bonus point.
And man-of-the-match Wilson was thankful for a big shift in the second-half from Ulster, that saw them move level on points with Glasgow Warriors at the GUINNESS PRO12 summit after three wins from their last three games.
"At half time we spoke about our discipline, which was better from last week, and about the breakdown, which was much better," he told the Belfast newsletter. "We upped the intensity in the second half and got our rewards off it.
"Our scrum went well and it has been a big weapon for us all year and the pack worked their socks off and put a lot of effort into it.
"Scarlets have a pretty tricky scrum and we had a few issues with it in the game over there at the start of the season. It certainly was a big weapon for us tonight.
"It was disappointing not to get the bonus point because we had massive momentum in the game in the second half. We had a couple of opportunities and could not finish them off.
"But at the same time we have got the win and that is important and we are still happy enough to have done that.
"To let them score at the end was also a bit disappointing. I think though at that stage of the game we had been putting it to them in their 22 and we were exhausted. Credit to them, they took their score well."
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