Barrow was on target with a chance midway through the first half only for team-mate Jay Fulton to end up holding his head in his hands after blocking the effort inside the six-yard box.
It took the Premier League side 34 minutes to break the deadlock against the side who lie one place off the League Two relegation zone.
Carroll set up Dyer’s chance and after keeper Owain Fon Williams blocked the first attempt, Dyer made no mistake with the second attempt.
Swansea’s second was created by Gambian recruit Barrow and finished stylishly by Carroll, who is on loan for the season from Tottenham Hotspur.
Barrow tapped in the third from close range after Gomis sent the ball across the face of the goal.
Swansea went on to win 6- 2 with Barrow and Gomis hailed as heroes of the night.
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