The visitors led 17-12 at the break and got off to a promising start, leading inside seven minutes.
Brett Connon kicked a penalty to the left corner, and a spell of patient phase play ended with a pick-and-go try down the opposite side from Scottish prop Murray McCallum.
Enthusiastic blitz defence from Elliott Obatoyinbo nipped a Bears attack in the bud as Newcastle aimed to cut off Bristol’s oxygen supply, and the visitors doubled their lead on 15 minutes when another penalty in the left corner ended with a try down the right.
This time it was Oli Spencer adding the finishing touch – the winger diving athletically inside the corner post after a ruck in front of the posts saw the ball saw the ball flung out to the width.
The home side got back into the contest on the quarter mark when a slick backs move saw Wales flyer Louis Rees-Zammit dart through a hole for an unconverted try down the right, and they led shortly after when ex-Newcastle centre Matias Moroni cut infield to score against his former club.
Tom Jordan’s conversion nudged them into a two-point lead, but a third penalty in the left corner delivered familiar returns for Newcastle when Ollie Leatherbarrow hit a hard midfield line to cross for a converted try between the posts.
That left his side 17-12 up at the break, and the Red Bulls added a further three to their lead when Connon chipped over a simple penalty from in front.
The Bears responded when Noah Heward sped down the right for an unconverted try, but Newcastle secured their four-try bonus point on 48 minutes when a short-side dart saw play worked out to Spencer, who scorched into the left corner.
Connon slotted the touchline conversion, but the see-saw scoring pattern continued when Kalaveti crossed in the left corner for Bristol’s bonus-point try.
Rees-Zammit put them into the lead just before the hour mark when he straightened the line and powered home from 20 metres, and they began to build a buffer when Gabriel Oghre scored from a driving maul in the left corner.
Joe Batley’s 20-minute red card threatened to tip the balance back in Newcastle’s direction, but the home side had enough to hold on for the win.
Bristol Bears: 15 Louis Rees-Zammit, 14 Noah Heward, 13 Matias Moroni (Rich Lane, 61), 12 James Williams, 11 Kalaveti Ravouvou, 10 Tom Jordan, 9 Harry Randall (Sam Wolstenholme, 56); 1 Max Lahiff (Ellis Genge, 43), 2 Harry Thacker, Gabriel Oghre, 43), 3 Lovejoy Chawatama, 4 George Taylor (Pedro Rubiolo, 43), 5 Joe Batley, 6 Joe Owen, 7 Fitz Harding (captain), 8 Viliame Mata.
Scorers - Tries: Louis Rees-Zammit 2, Matias Moroni, Noah Heward, Kalaveti Ravouvou, Gabriel Oghre. Conversions: Tom Jordan 2, James Williams.
Newcastle Red Bulls: 15 Ethan Grayson, 14 Oli Spencer, 13 Alex Hearle, 12 Sammy Arnold (Cameron Hutchison, 66), 11 Elliott Obatoyinbo, 10 Brett Connon (Boeta Chamberlain, 66), 9 Simón Benítez Cruz (James Elliott, 66); 1 Adam Brocklebank (Connor Hancock, 58), 2 George McGuigan (captain, Ollie Fletcher, 53), 3 Murray McCallum (Richard Palframan, 54), 4 Oscar Usher, 5 Seb de Chaves (Finn Baker, 66), 6 Tom Gordon (Amanaki Mafi, 58), 7 Tom Christie, 8 Ollie Leatherbarrow.
Scorers - Tries: Murray McCallum, Oli Spencer 2, Ollie Leatherbarrow. Conversions: Brett Connon 2.
Referee: George Selwood.
Attendance: 22,003.