The match started in frantic fashion and Price passed up a chance to strike early for the hosts when he hacked ahead a loose ball, but thinking he had put too much on it he momentarily gave up the chase. The bounce kept the ball in the in-goal area but that hesitation allowed Finn Russell to nip in ahead of Price and touch down for a goalline dropout.
Thompson then sent a fairly straightforward penalty attempt crashing off the post and moments later Bath struck the first blow when Underhill crashed over from close range.
The visitors looked like they were settling into their stride until they were caught out by a quick line-out, which led to Tuipulotu going over and Thompson converting to level things up.
The home side were having to do most of the defending and the Bath power started to show. They were gaining ascendency in the scrum and hooker Dunn regained the lead when he dotted down after a strong rolling line-out maul.
Thompson kicked a penalty to keep Edinburgh within two at the break and they got their noses in front early in the second half.
They engineered an overlap out wide and Wes Goosen looked set to put Darcy Graham over in the corner until a cynical knock-on by Tom de Glanville. A yellow card for De Glanville, a penalty try and Edinburgh led 17-12.
Bath went back down the other end and hammered at the Edinburgh line. They went over through Beno Obano but a television match official (TMO) intervention showed he had been held up and the try was ruled out.
The home side were under immense pressure and it was only a matter of time before their defence was breached. Dunn eventually went over for his second try, a carbon copy of his first after a powerful line-out maul.
The Premiership leaders were well on top now and substitute Barbeary dived over for their fourth try to give them a one-score cushion on the scoreboard.
Russell kicked three points to put Bath 10 points to the good, but just as the match appeared to be drifting away from the home side, they struck with a cracking try.
Thompson ghosted through a gap in the midfield and found Price running a great support line to beat the cover defence and scored to bring his side back within three.
The push to snatch a dramatic victory did not materialise from Edinburgh and Bath ultimately made their superiority count with late tries from substitute Annett and back-row Pepper to seal a deserved victory.
