The hosts were 13-7 up at the interval with a try from David Lemi which was converted by Jason Strange who added two penalties before the break.
Newcastle's response came in the form of a try from full-back Matthew Burke with Toby Flood adding the conversion.
After Strange kicked his third penalty in the match, he added the conversion to Lemi's second try of the afternoon.
Newcastle director of rugby Rob Andrew:
"We felt comfortable at half-time being just six points behind and we thought the game was ours for the taking having coped well in the first half scoring a try into the wind. "Unfortunately Bristol's second try came from a clear forward pass, it is not a gripe but a fact.
"It was about two yards forward which everyone saw except the referee, but never mind.
"That then allowed Bristol to dictate the pace of the last quarter which they did extremely well - though not a lot happened in that spell.
"I didn't think there was a lot to choose between the two teams."
Bristol: Going; L Robinson, Higgitt, Cox, Lima; Strange, Perry; Hilton, Regan, Crompton; Budgett, Llewellyn; Salter (capt), El Abd, Ward-Smith.
Replacements: Irish, Nelson, Winters, Sambucetti, Short/Lewis, Gray, Lemi.
Newcastle: Burke; May, Noon, Mayerhofler, Shaw; Flood, Charlton; Ward, Long, Morris; Perry, Buist; McCarthy, Harris, Charvis (capt).
Replacements: Williams, Thompson, Finegan, Woods,Grindal, Walder, Elliott.