Bath dominated the early exchanges and Vesty claimed a try after quick hands created a gap in the Irish line.
Moments earlier Delon Armitage had been handed a yellow card for a high tackle on Tom Biggs.
Bath led 5-0 at half-time but Irish started the second half strongly and Tom Homer landed two penalties to edge them ahead 6-5.
Bath regained the lead when Dave Attwood went over in the corner for their second try which was viewed by the video referee before being awarded.
Homer booted over his third and fourth penalties to give Irish a 12-10 lead and it was beginning to look as though Irish would rue their missed conversions.
But Vesty had the final word with the penalty in the closing minutes following an infringement by Shontayne Hape.
In-form Irish had gone into the match looking for a fifth win in six Aviva Premiership outings.
Bath's win leaves them in fourth place with Irish in sixth.
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