The Welsh side were in pole position in Galway leading in the second period before the hosts fought back to take the victory and continue their unbeaten start to the season.
The Scarlets had all the early possession and Dan Newton kicked them ahead with an eighth-minute penalty.
Both sides tested each others defences but both held firm for the opening forty minutes. Newton and Matthew Jarvis exchanged penalties just before the break to give the visitors a 6 v 3 lead at half-time.
Flanker Rob McCusker scored the game’s first try just five minutes in the second period. He crashed over after a close range scrum and Newton added the extras.
Mark McCrea then pulled a try back for the hosts and Jarvis converted to bring the scores to within just a point.
Jarvis’ second penalty ten minutes from time gave Connacht the lead for the first time in the game and they hung on to take the victory.
