"The way we played without the ball was the difference, our defence was excellent," said Llanelli head of rugby Anthony Buchanan.
"It was always going to be a close game. Credit to Quins, they came down and looked to attack us.
"There is a lot to work on, but it's the start of the season. We will look forward to Llandovery on Saturday now."
Llanelli had led 16-9 at half-time though the Quins will have felt hard done by having had the better of the first half.
The opening 40 minutes had predominantly been a kicking contest, fly-halves Josh Lewis and Craig Evans kicking three penalties apiece to make the score 9-9 with precious first half seconds remaining.
But in what proved to be the crucial moment of the match, Llanelli launched a sparkling attack from deep through full-back Dion Jones, a break that led to Josh Adams cutting inside to score.
The Quins reduced the arrears with a fourth Evans penalty shortly after the restart before Llanelli were denied a second try when referee Nigel Owens ruled out a Kyle Evans score.
Owens then yellow carded Llanelli flanker Dan Thomas for an infringement at a ruck and although Quins pressed late on, a penalty from replacement full-back Aled Thomas was all they could manage in reply.