The visitors led 28-14 at half-time and were ahead inside seven minutes when Connor Slevin scored in the right corner to exploit an overlap – the fly-half converting his own try from wide out.
The sides were back level again when a midfield turnover from James Harper and a quick transfer to the width freed Obi Ene for a break down the right, his inside ball finding Ethan Grayson for the try.
Teenage fly-half Jake Dalziel curled in the touchline conversion, but the Londoners led once more on the quarter mark when Nick David supported a line-break and collected a simple pop pass for a converted try.
Grayson levelled proceedings again just after the half-hour when super support play from James Elliott’s line-break saw him collecting a scoring pass from Adam Scott.
But the Londoners responded almost instantly when No.8 Zach Carr collected an inside pop to score between the posts.
Quins grabbed their four-try bonus point just before half-time when hooker Jimmy Staples scored in the right corner from simple hands along the line, Slevin stroking over the conversion for a 14-point interval advantage.
The Red Bulls began the second half with no shortage of attacking ambition, coming just a whisker away from a length-of-the-field try when a scrum inside their own 22 saw Alex Hearle breaking the line and feeding Nathan Greenwood for a downfield burst.
But Harlequins were more clinical in finishing off their opportunities, Cameron Anderson registering their fifth try of the day when the winger plotted a winding course to the line from flat-foot ball 45 metres out.
David followed up with number six when the winger hit a well-timed diagonal run to the right corner for an unconverted score, but the Red Bulls responded with two tries in quick succession.
The first came from a penalty try as an advancing maul in the left corner was illegally pulled down, and that was rapidly followed by an Alex Hearle score down the same side of the field as the home team worked the ball through the hands to the space.
But it was too little too late for the home side, with Quins grabbing one last try through Sean Sharp before a consolation try with the final play from lock Rob Carmichael.
Newcastle Red Bulls: 15 Nathan Greenwood, 14 Obi Ene (Oli Spencer, 57), 13 Ethan Grayson, 12 Rhys Beeckmans, 11 Alex Hearle, 10 Jake Dalziel, 9 James Elliott (Joe Davis, 57); 1 Tom West (Adam Brocklebank, 51), 2 Ollie Fletcher (Samson Adejimi, 51), 3 James Harper (Murray McCallum, 51), 4 Franco Molina (Rob Carmichael, 40), 5 Finn Baker, 6 George Bolam, 7 Mason Flesch (Adam Scott, 58), 8 Sam Graham (captain, Tom Christie, 66).
Scorers – Tries: Ethan Grayson 2, Penalty try, Alex Hearle, Rob Carmichael. Conversions: Jake Dalziel 3.
Harlequins: 15 Josh Bellamy, 14 Nick David, 13 Oscar Beard (Sean Kerr, 51), 12 Ben Waghorn, 11 Cameron Anderson (Sean Sharp, 58), 10 Connor Slevin, 9 Stu Townsend; 1 Harry Johnson-Holmes (Ethan Clarke, 51), 2 Jimmy Staples (Jack Musk, 58), 3 Harry Williams (Ruan Swiegers, 58), 4 Elliot Williams, 5 Harry Browne, 6 Tate Williams, 7 Will Evans (captain, Leo Dickinson, 74), 8 Zach Carr.
Scorers – Tries: Connor Slevin, Nick David 2, Zach Carr, Jimmy Staples, Cameron Anderson, Sean Sharp. Conversions: Connor Slevin 5.
Referee: Harry Walbaum. Attendance: 5,247.