Leinster got off to a brilliant start with a try by full-back Andrew Conway after a good line break from inside-centre Andrew Goodman, who went over minutes later to make it 10-0 to the visitors.
Despite Leinster’s dominance in the half’s early stages, they only had a point advantage going in at the break with the score at 10-9 to the visitors.
After Mariano Sambucetti went over right under the posts, Adrian Jarvis duly converted to put Bristol ahead for the first time in the match. But Leinster reacted immediately as James Tracy went over after a series of missed tackles from the English side.
With five minutes remaining, winger Fionn Carr clinched what proved to be the winning try after a good break from Reid, who kicked over the subsequent conversion. But it was the home side who had the last points however with a rolling maul eventually forced over the line by replacement hooker Ross Johnston.
Leinster head coach Girvan Dempsey was delighted with his team after the final whistle.
“We knew Bristol were a strong side and have had some good wins in the competition this year. I’m delighted with the way the guys fronted up.
“I was very impressed with the forwards and young Rhys Ruddock led really well from the front.”
BRISTOL: J Goodridge; G Watkins, F Otto, B Mosses (J Tovey, 26), A Short; A Jarvis (T Roberts, 78); C Braley (J Grindal, 62); M Lilley, R Lawrence (R Johnston, 65), G Cortes (B Douglas, 44), M Sambucetti (G Townson, 65), R Winters (capt), A Faosiliva (M Mama, 58), J Merriman, M Eadie.
Replacements: R Johnston, B Douglas, G Townson, M Mama, J Grindal, T Roberts, J Tovey.
LEINSTER A: A Conway; A Boyle (S Coghlan-Murray, 70), N Reid, A Goodman, F Carr; C Marsh (C O’Shea, 73); L McGrath (J Cooney, 62); J Tracy, A Dundon (T Sexton, 67), M Bent, T Denton (T Beirne, 56), M Flanagan, B Marshall, R Ruddock (capt) (D Kenny, 76), L Auva’a.
Replacements: T Sexton, M Moore, T Beirne, D Kenny, J Cooney, C O’Shea, S Coghlan-Murray Referee: Andrew McMenemey.