A try from Alapati Leiua and 12 points from the boot of Ian Madigan saw them beat a disappointing Bath side, who made far too many errors.
This victory delighted the 26,300 supporters in the ground as Bristol’s comeback to the top flight of English rugby kicked off in real style.
Bristol were 6-0 in front at the break despite being on the back foot for a large spell of the opening period, with Bath wasteful in possession.
Madigan set the Premiership new-boys on their way with a ninth minute penalty before Priestland hit the post with a relatively simple shot.
Bath would then enjoy a sustained period of ball and it looked like Tom Homer had been put over by Priestland on 15 minutes for an easy dot down. However, replays showed the full-back had lost the ball over the line under no pressure whatsoever, with the five points chalked off.
Still Bath came though with more ball but this time it was Jordan Crane’s impressive work at the breakdown that helped relieve the pressure.
Bristol used that as a catalyst and Luke Morahan’s break on halfway should have resulted in points. However, the wing could not get the pass out to Nic Stirzaker and was instead hauled down two metres from the whitewash. At least for Bristol two minutes later Madigan would make it 6-0.
6-0 became 9-0 on 43 minutes after Sam Underhill was pinged at the breakdown, Madigan making no mistake. And with Priestland wayward once again off the tee seven minutes later, confidence was growing in Bristol.
But Bath would not roll over and finally Homer had a chance to right the wrong of the first-half when he finished off a slick move to make it 9-7.
Bath were now on top and with Tusi Pisi sin-binned after coming offside a metre from his try-line, the visitors were pressing for a crossing.
It would not come though and they had to settle for three points from 35 metres from Priestland. However, Madigan responded to make it 12-10.
That seemed to galvanise the Bears and Morahan’s break on 72 minutes saw him find a hole and then offload to Leiua, who juggled before gathering for the crucial try. Madigan could not add the extras which meant Bath remained just one score behind at 17-10 with eight minutes remaining.
The visitors pushed hard at the death for that levelling seven points but were unsuccessful, with a crooked line-out five metres out summing up their dismal evening as Bristol celebrated a famous win.
Pens: Madigan 4
Yellow Card: Pisi