Mark Regan went over after 10 minutes to put the dominant visitors ahead, but Tom Rees surged past Jake Rauluni and David Lemi to score in the corner.
After the break Alex King kicked a second penalty, and Tom Voyce began a 70-metre run to the line, beating a host of players to put Wasps ahead.
King converted and though Jason Strange reduced the gap to two points after 64 minutes another King penalty sealed it.
Bristol coach Richard Hill:
"The players are very disappointed they haven't won the game. "We've got a number of players who haven't played at this level and they were concerned about coming here, but they showed a lot of composure."
Wasps: Brooks, Sackey, Erinle, Abbott, Voyce, King, Reddan, Payne, Ibanez, J. Dawson, Skivington, R. Birkett, Hart, Rees, Dallaglio.
Replacements: Hoadley for Brooks (73), Gotting for Ibanez (41), Bracken for J. Dawson (48).
Not Used: Lock, Chamberlain, Honeyben, Staunton.
Bristol: Going, Stanojevic, Higgitt, Cox, Lemi, Strange, Rauluni, Hilton, Regan, Crompton, Winters, Llewellyn, Salter, El Abd, Lewis.
Replacements: Denney for Cox (41), A. Clarke for Hilton (62).
Not Used: N. Clarke, Kohn, Martin-Redman, Teague, Robinson.
Sin Bin: Cox (30).
Att: 7,901
Referee: David Rose (RFU)
MMIX
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