Gleeson was carried off on a stretcher with a worrying injury late on and by then, Munster ‘s hopes of victory at the Memorial Ground had long gone.
A hatful of wasted chances and two penalty tries in the space of eight minutes left Munster depending on other results to reach the quarter-finals of this competition.
Yet coach Anthony Foley will have been left scratching his head at how his team trailed 6-3 at half-time having dominated virtually every aspect of the half
The only period of genuine concern came in the opening minutes but escaped a series of penalties following a powerful Bristol driving lineout.
After that it was simply one red wave after another with Munster looking far more dynamic and threatening.
Fly-half Cusack gave an assured display, opening his account in the seventh minute and then almost created a try with his kick to Sean Scanlon, but the wing failed to find the on-rushing Scott Deasy with his pass.
However, Munster were guilty of lacking composure at the vital moment and time again, the attack broke down within sniffing distance of the try line.
Captain Billy Holland was among those guilty after Cusack pushed his second effort wide.
Of concern for Foley was Munster ‘s scrum that struggled to get out of reverse against the English eight and allowed Barnes to put Bristol ahead before half-time with a second penalty in three minutes.
Bristol flanker James Merriman but Munster failed to capitalise in the face of Bristol ‘s power and after No8 Paddy Butler followed to the sin bin, the front-row capitulated a penalty try.
Just moments later and another lineout drive led referee Sean Brickell to award a second try and show Dave Foley a yellow card. Barnes converted both and the game was gone for Munster .
Gleeson left the field late on before Bristol ‘s Anthony Elliott scored an interception try and Munster ‘s Danny Barnes grabbed a late consolation.
Bristol scorers: Tries: Penalty 2, Elliott. Cons: Barnes 3. Pens: Barnes 2.BRISTOL: J Tovey; A Elliott, J Adams, J Fatialofa, G Watkins; E Barnes, R Bolt (S Alford, 67); M Irish (J Hobson, 60), R Johnston (D Blaney, 73), D Compton (W Thompson, 63), D Barry (M Sambucetti, 60), R Winters, D Montagu (J Miller, 67), I Grieve (capt),.
Munster scorers: Tries: Barnes. Cons: Cusack. Pens: Cusack.
MUNSTER A: S Deasy (R O’Mahoney, 67); D Barnes, T Gleeson (S Deasy, 73), T Smith (P Rowley, 72), S Scanlon (K Essex, 52, C Hircock, 62); D Cusack, C Murray; D Kilcoyne (D Hurley, 50), D Casey (G Slattery, 60), P Borlase, B Holland (capt), D Foley, D O’Callghan (B Hayes, 77), P Butler, B O’Hara (K Essex, 62).
Referee: Sean Brickwell (Wales)