The visitors led 31-7 at the break and drew first blood on five minutes when Paul Brown-Bampoe crossed for an unconverted try in the right corner – the winger benefiting from quick hands along the line after a penalty was kicked to touch.
Only a jackal turnover from Richard Palframan denied Exeter a second try when the prop thwarted a close-range foray, but there was no stopping them on ten minutes when the ball was flung wide to Campbell Ridl in the left corner.
Harvey Skinner this time added the conversion, and the Devon side were three tries up before the quarter mark when the ball was worked to the blindside from the back of a driving maul, Brown-Bampoe scoring his second of the evening.
Exeter wrapped up their bonus point on 25 minutes when full-back Olly Woodburn hit a hard midfield line and powered over, but the Red Bulls got on the board just after the half-hour.
Christian Wade was the man to make it happen, the winger receiving a long cut-out pass from Simón Benítez Cruz and showing pace and power to carve a path to the try-line through a heap of traffic.
Brett Connon added the conversion, but Exeter seized back the momentum when Ridl darted in for his second try of the evening, collecting an offload from Woodburn in the left corner.
That put the Chiefs 31-7 up at the break, and they were back in the points just two minutes after the resumption when Italy scrum-half Stephen Varney glided over from a line-out strike launch.
The home side kept probing and almost had their second try when a succession of red-zone penalties saw Adam Brocklebank going close in the left corner, the prop being driven into touch just a metre short of the corner post.
The Red Bulls turned to their bench in search of impetus – loan signing Ben Healy coming on for his debut and Oli Spencer making his 50th appearance for the club.
The two combined almost instantly when Spencer made a line-break into the Chiefs’ 22 and Healy fired a cross-field kick from the next phase, but the bounce of the ball eluded the onrushing Wade on a night of frustration.
There was at least a late moment of consolation when replacement Connor Hancock scored from a pick and go with the final play, Healy converting.
Newcastle Red Bulls: 15 Elliott Obatoyinbo, 14 Christian Wade, 13 Alex Hearle, 12 Rhys Beeckmans, 11 Harrison Obatoyinbo (Oli Spencer, 57), 10 Brett Connon (Ben Healy, 54), 9 Simón Benítez Cruz (James Elliott, 75); 1 Adam Brocklebank, 2 George McGuigan (captain, Ollie Fletcher, 58), 3 Richard Palframan (Murray McCallum, 40), 4 Oscar Usher (Connor Hancock, 68), 5 Adam Scott (Freddie Clarke, 48), 6 Reuben Parsons (Tom Gordon, 48), 7 Tom Christie, 8 Ollie Leatherbarrow.
Scorers - Tries: Christian Wade, Connor Hancock. Conversions: Brett Connon, Ben Healy.
Exeter Chiefs: 15 Olly Woodburn (Dan John, 71), 14 Paul Brown-Bampoe, 13 Nick Lilley, 12 Will Rigg, 11 Campbell Ridl, 10 Harvey Skinner (Will Haydon-Wood, 60), 9 Stephen Varney (Tom Cairns, 60); 1 Scott Sio (Ethan Burger, 54), 2 Jack Yeandle (captain, Julian Heaven, 54), 3 Jimmy Roots (Bachuki Tchumbadze, 54), 4 Rusi Tuima, 5 Andrea Zambonin (Lewis Pearson, 60), 6 Tom Hooper (Finn Worsley Brady, 60), 7 Ross Vintcent, 8 Greg Fisilau.
Scorers - Tries: Paul Brown-Bampoe 2, Campbell Ridl 2, Olly Woodburn, Stephen Varney. Conversions: Harvey Skinner 4.
Referee: Craig Maxwell-Keys. Attendance: 6,080.