Jordan Williams was prolific on kicking duty in the opening period, firing five penalties through the uprights for Scarlets.
However, the home side found themselves down to 13 men for a time, with Emyr Phillips and Andy Fenby sent to the sin bin.
Tries from Thomas Waldrom, Andy Forsyth and a penalty try saw Tigers take a narrow lead at half time in a closely-matched opening 40 minutes.
It was the visitors who let their discipline slip after the break, as Rob Andrew and Fraser Balmain were both shown yellow cards within the space of four minutes.
Williams kicked the Welsh side back in to the lead, before Fenby took advantage of a mix-up in the rook from Leicester to score Scarlet's first try of the game inside the last 10 minutes, before Gareth Owen and Nic Reynolds scored their second and third respectively just before full time.
The result sees Leicester finish second in pool two, while Scarlets climb to third in pool three.