The home side were indebted to tries from wing Andy Fenby, full-back Gareth Owen and centre Nic Reynolds in the final 10 minutes which backed up seven penalties from outside-half Owain Williams.
But Leicester, who led 19-15 at half-time, were not helped by losing forwards Fraser Balmain and Rob Andrew to the sin-bin in the second half.
Scarlets began brightly with Williams giving them good territory with a kick to the corner.
From that position the Tigers defence was given a stern test until they conceded a penalty on seven minutes which Williams duly converted.
But that lead was shortlived as a solid drive from a line-out saw number eight Thomas Waldrom emerge from a pile of bodies as the try scorer. Centre Matt Cornwell failed to added the extras.
Scarlets hit straight back as Williams put over two penalties in the space of four minutes, including one from 45 metres.
Williams made it 12-5 on 22 minutes with his fourth shot at goal after the Scarlets had gone close to scoring a try with runs from Fenby and debutant Kyle Evans.
However, that purple patch ended as captain Emyr Phillips and Fenby were issued yellow cards in the space of two minutes. Phillips went to the bin for going in from the side attempting to stop a Tigers drive to the line, while Fenby was penalised for a deliberate knock-on.
Down to 13 men, it was no surprise that Scarlets conceded a penalty try after collapsing a scrum. Cornwell added the extras as the scores were levelled at 12-12.
A fifth penalty from Williams made it 15-12 on 35 minutes but with Phillips and Fenby back on the field, Scarlets conceded a try in injury time as centre Andy Forsyth ran in down the left after profiting from an overlap.
The home side's positive start to the second half was rewarded with Williams' sixth penalty as the Tigers lead was cut to one point.
Williams saw another penalty come back off the post but converted his seventh penalty as Scarlets regained the lead at 21-19 on the hour mark just before lock Andrew was sin-binned for tipping at a line-out.
And as the Tigers scrum continued to struggle Balmain found himself in the bin with Andrew.
With two extra men Scarlets celebrated their first try when Fenby ran away to score down the left to make it 28-19.
But they had not finished there with Owen and Reynolds going over for two more tries in the final three minutes.
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