Wasps had the opening weekend off but ensured it was worth the wait for the crowd inside the CBS Arena as some clinical finishing and superb defence helped them dispatch last year's semi-finalists.
They did have captain Brad Shields, another of their try scorers, sent off in the closing stages, but by that point the result was in the bag against a Bristol side who have now lost both of their fixtures this season.
Callum Sheedy kicked the first points of the match for Bristol after just three minutes, although Jacob Umaga should have cancelled this out as he sent a penalty of his own wide from a central position.
The Bears then lost hooker Harry Thacker to injury with eight minutes gone and suffered a further setback when Wasps clicked for the opening try.
It came from Umaga doing well to wriggle out of a tight area before passing the ball left for Crossdale, one of four players in the hosts' line-up making their debut, who raced away to score in the corner.
Umaga could not add the conversion, but it was third time lucky from the tee for the fly-half when he sent over a penalty to increase his side's lead to 8-3.
Shields was then sin-binned after the hosts conceded a succession of penalties, and Bristol pressure eventually told as prop Max Lahiff bundled over from close range to level the match.
Parity did not last long, however, as Umaga's second penalty quickly nudged Wasps back in front and the gap was extended to eight points four minutes before half-time.
Dan Robson spotted Crossdale in space on the right and threw a terrific pass that sent the full-back through a gap for a fairly simple finish.
Wasps were able to pick up where they left off at the start of the second half, as Shields managed to rumble his way over with the help of a couple of his fellow forwards.
The match was over as a contest in the 64th minute. Replacement hooker Dan Frost, another Wasps new boy, picked up the loose ball following a turnover and sprinted clear from 40 metres.
Shields was sent off shortly afterwards, but Wasps still had two more tries in them, with Tom Willis going clear from Josh Bassett's offload before Zach Kibirige took advantage of a Bristol fumble off the final play.