After an early penalty from Scarlets' Steve Shingler, the game became a physical contest which forced the hosts to make three substitutes in the opening half an hour.
Dorrian converted the try, and then added his second penalty to give the visitors a 13-6 lead at half time.
Scarlets attacked in the second half, and their pressure was rewarded in the 54th minute when Harry Robinson scrambled across for their first try.
They grew in confidence as the game went on and took the lead when Rhodri Williams exploited a gap in the Exiles defence to cross for their second score.
The Welsh side then confirmed victory late on when Robinson latched onto the end of a clever kick to set up Kyle Evans to score.
Irish pulled a late try back through Theo Brophy-Clewes, but it was not enough to stop Scarlets winning the match.