A first half score from Frazer Climo on his Pro12 debut gave Simon Easterby’s men a 10-9 half time lead.
The start of the second half saw the visitors pull further clear and when Kirby Myhill and Gareth Davies crossed for two quick scores, the game looked sewn up at 24-9.
But late on, tries from Matt Healy and Jason Harris-Wright left the Scarlets sweating.
The game had started with Connacht moving into a 6-0 lead and although Aled Thomas got things moving for the Scarlets, pressure from the home side led to Jacobie Adriaanse receiving a yellow card.
The Irish province failed to make the most of their man advantage though and during the time the Scarlets were down to 14, they scored just three points.
And when Climo crossed the whitewash, Thomas converting, the Scarlets led by a point at the break.
Tries from Myhill and scrum half Davies meant the Scarlets started with a bang after half time, Thomas adding the extras to both scores.
It looked to be all over but Connacht refused to lie down with Healy and Harris-Wright dotting down either side of a yellow card for Wales flanker Aaron Shingler.
Easterby’s side had their backs to the wall but they came through in the end.
Â
