Jack Wallace, Tom Varndell, Billy Searle and Luke Arscott scores sealed Bristol's first try-scoring bonus point of the season, putting them 22-7 up at half-time of the Challenge Cup game.
Chris Booker's brace and a Thretton Palamo try sealed Bristol's second win in all competitions this season.
In response Watisoni Votu and Thibault Daubagna crossed for Pau.
Acting head coach Mark Tainton guided the West County side to their first win since Connacht's Pat Lam was confirmed as the club's new boss for next season.
Before overcoming Pau, Bristol's last triumph in European competition came against Montpellier in the Challenge Cup in January 2009 - the same season that Bristol were relegated to the Championship, where they stayed for seven season.
Victory, just their second of 2016-17 after overcoming Sale Sharks in the Anglo Welsh Cup in November, moves Bristol off the bottom of Pool Four of the European Rugby Challenge Cup, above Top 14 club Pau and to within three points of near neighbours Bath in second.
Bristol: Wallace, Varndell, Palamo, Arscott, Lemi, Searle, Uren, Tonga'uiha, Crumpton, Perenise, Nemsadze, Evans, Koster, Fenton-Wells, J Phillips.
Replacements: Brooker, O'Connell, Ford-Robinson, Glynn, Joyce, Roberts, Jarvis, Tovey.
Pau: Malie, Votu, Fumat, Dupouy, Lestremau, Fajardo, Moa, Moise, L Rey, Hamadache, Tutaia, Boutaty, Dougall, Kueffner, Butler.
Replacements: Lespiaucq-Brettes, Jacquot, Tierney, Pesenti, Bernad, Tomas, Daubagna, Boundjema.
Referee: Dudley Phillips (Ireland)
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Bristol won in Europe for the first time in almost eight years, thrashing French side Pau at Ashton Gate.