Adams scored two tries on his league debut after signing from Gloucester
Bristol stumbled to their second consecutive league victory with an unconvincing away win at newly-promoted Birmingham and Solihull.
The vistors ran in six tries but some sloppy play and indiscipline allowed the Bees to close to within four points, making for a tense climax.
Bristol assistant coach John Brain told BBC Radio Bristol: "We were just loose and lethargic.
"We've got to get rid of that slackness and dissect why we felt flat."
He added: "When you come to places like this you've got to make sure your reserve tanks are full and you're ready to go on all cylinders and we weren't."
Bristol were unable to extend their lead by a convincing margin for the whole match and were thankful to their new signings for keeping their noses in front.
In addition to a penalty try, captain and scrum-half Jason Spice, winger Dan Norton and back-rower Dan Montagu all touched down.
The remaining two tries were scored by centre Jack Adams, who was making his league debut.
"We played wide-open rugby but we just weren't clinical with a lot of our stuff," said Adams.
"There were some sloppy penalties and they did punish us for it.
"That will all be sorted out on Monday. We'll all be in the meeting room going over it and I'm sure the boys will be a bit gutted with that."
Bristol sit in fourth place in the Championship, level on nine points with Exeter Chiefs, Cornish Pirates and Bedford Blues.
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