All the points came through penalties at Sandy Park.
It was a typical derby clash with little between the teams but the outcome meant a fourth home defeat for the Chiefs in the league this season, and the Devon side have to look back to January 1978 for when they last beat Bath.
Tom Heathcote, at just 20, reinforced his growing reputation with four penalties but backs and defence coach Davis believes the visitors could have won it more comfortably.
"It was close in terms of the scoreline but I think over the 80 minutes we deserved the result," he said. "We were the side trying to play the most rugby and I thought the foundation of the win was laid in our defence.
"The forwards, in the last 20 minutes especially, were outstanding in their driving play and the scrum which earned us the points that got us across the finish line.
"A lot of credit has to go to the players because their character was absolutely first rate. We created numerous line-breaks and a little lack of accuracy probably cost us a couple of tries but we were good value for the win.
"We tried to go with tempo and speed from the start of the match but in the first 40 minutes we came up with some unforced errors because the game got a bit fractured and it wasn't structured.
"We spoke at half-time and just that we just wanted a bit more accuracy, obviously we thought we had a slight breeze, and we wanted to keep up the tempo because we felt we were getting the edge.
"The guys did that and had a bit more patience on the ball which put Exeter under pressure and we felt that if we could do that then the penalties would come and that is how it panned out."
Academy half-back Heathcote, a full-time student at the University of Bath who has earned a new three-year contract, took over the kicking duties after centre Olly Barkley had been unsuccessful with two attempts.
Argentinian fly-half Ignacio Mieres kicked three penalties for the Chiefs, who missed out on going in to fourth place in the table having to settle instead for a losing bonus point.
Exeter have now lost at home four times this season and Chiefs head coach Rob Baxter says that they will have to perform better next week when they host Sale Sharks.
He said: "I was disappointed. Not disappointed that we lost a game of rugby because we have learned to deal with that over the years, it is disappointing that we didn't play as well as we could.
"I am not taking anything away from Bath, they probably feel the same, but I think we made too many mistakes, we coughed-up too much soft ball and gave away too many soft penalties to say we are happy.
"Sometimes you lose games and it is just tough but you have played well and given it everything but that hasn't happened in this game but it is not the end of the world. It is a bit of a lesson for us as we go along."
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