The Irish Province were the only Celtic League side to win at Stradey last season, and their repeat performance hoists them into third place, two points ahead of Leinster who play on Sunday.
With the majority of territory and possession in the first half the Scarlets were able to build their early lead through a Gareth Bowen penalty and a Dwayne Peel try, which Bowen converted. However Connacht defended with vigour and repelled a wave of Scarlets attacks before Conor O'Loughlin was able to dart over in one of their first forays into the home side's 22, with Paul Warwick converting.
The visitors moved into the lead early in the second period when Ted Robinson crossed and Warwick converted. A Conor McPhillips try further increased the margin, but Bowen reduced the gap with a second penalty and the Scarlets regained the lead 10 minutes from the end when Dafydd Jones appeared from the bottom of a pile of bodies to claim the push-over try.
However Connacht came straight back and within five minutes Mark McHugh had stroked over a penalty from half way. The game was sealed by an injury time try for Michael Swift which also secured the bonus point.