Despite losing several players through injury, the Welsh region scores tries in each half through Harry Robinson and Aled Davies to give themselves a great chance of progressing from pool three.
The defeat is a massive blow to the Tigers hopes of reaching the quarter-finals with a daunting double-header against Toulon to come in rounds three and four.
Tigers should have been in front early on when fit-again centre Anthony Allen fumbled the ball over the line after 10 minutes with the try-line at his mercy.
Former Scarlets fly-half Owen Williams finally broke the deadlock for the visitors with a penalty but it proved to be their only points of the evening.
The Scarlets were gifted the game’s opening try after 17 minutes when Tigers wing Miles Benjamin dropped the ball inside his own 22 and opposite number Robinson went over in the corner.
Rhys Priestland missed the conversion and two penalties before Steve Shingler took over the kicking duties and slotted three points with the boot.
But Shingler and Tigers’ Williams both missed chances before the break to leave Scarlets leading but just five points at the interval.
When the Scarlets did eventually get their second try it once again came gift-wrapped from the Tigers. Ben Youngs tried to go outside opposite number Davies and when the England international attempted to flick the ball out of the tackle, Davies stole possession and sprinted clear to score.
Shingler added the conversion to seal an historic win for the Scarlets and give them more than a fighting chance of becoming the first Welsh region to reach the quarter-finals since 2011.
