Wing Tom James took the move on and then Andy Powell was dragged down yards short of the Bath line.
Two penalties from Glen Jackson reduced the gap but a Davis penalty and a Peter Short try ensured Bath were on top.
Barkley's fifth penalty gave Bath a short-lived 18-16 lead, before Everitt restored the Irish lead at 19-18 with a penalty a minute later.
But it left Bath two points short of a place in the final.
BBC: SPORT | Rugby Union | European Club | Montferrand 38-22 Bath
But he was on target two minutes later from a penalty which saw Bath lock Peter Short sent to the sin bin for pulling back Blues number eight Xavier Rush off the ball.
In contrast, Ackman is in line for quite a bath on his Herbalife short today, and by the way good old MBIA is on quite a run of its own.
Dylan Hartley was held just short of the line on 19 minutes but when Bath prop David Flatman was penalised at the resulting scrum, Myler made it 10-6 with his second penalty, only for James to extend his side's lead after the visitors were offside at a ruck.
Supermarine play in the Southern League Premier Division, two levels below Bath, and a win would be the biggest in their short history.
Academy graduate Nick Scott came in at short notice for Matt Banahan, who was not released by England, to make his first start for Bath.
Bath began the second half on the front foot but were punished for turning the ball over when Short hacked on Farrell's punt upfield and Ratouvou touched down.
The former Bath coach had previously had two spells on England's backroom team as a forwards coach after a short stint as head coach of Ireland.
Bath's Nick Abendanon was also drafted into the squad in place of Tom Varndell, who is short of match practice after falling out of favour at Leicester.
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