Gary Johnson's men had thrown away a 3-1 lead at Ashton Gate but they proved they were the better side over two games.
Scott Murray and Enoch Showunmi were the heroes of the night and Bristol could have won the game by a wider margin.
Adams made just two changes following the calamitous performance against Crystal Palace, whereas Bristol kept faith with the side which came back so well against Tranmere.
Johnson will have been pleased with the way his side started the game, although his son Lee might have earned himself a rollicking.
Showunmi set up the midfielder in the seventh minute but he fired over when really he should have at least hit the target.
Phil Jevons also saw a sight of goal early on but failed to take advantage before Showunmi had the ball nicked off him by Andrew Whing just as he closed in for a header inside the six-yard area.
But it was far from one sided as Coventry went close on several occasions without ever really convincing.
Kevin Kyle shot over Adriano Basso's bar when Jay Tabb set him up in the 18th minute while Stern John also kept the keeper busy.
It was, however, the Robins who had the slight edge and Coventry failed to pick up the hints.
On three occasions Johnson's men took corners or free-kicks to the near post but their finishing let them down.
The visitors also found considerable joy just outside the 18-yard area as Coventry sat deeper.
It eventually cost them seven minutes from the interval when Showunmi fed Murray and he neatly curled a right-foot shot around Luke Steele and into the net.
Adams decided to take off Isaac Osbourne and bring on the more attack-minded Kevin Thornton.
But the Irishman had little chance to influence the game before Bristol carved Coventry open for their second and deserved goal.
Thornton and Michael Doyle made a hash of things just outside the area and presented Johnson with possession. He looked up and then delivered the perfect cross for Showunmi who soared above Robert Page and headed powerfully down into the net.
Adams immediately made a double substitution bringing on Dele Adebola and Chris Birchall in a bid to turn things around.
But despite that it was League One side who still made the running. Steve Brooker twice tested Steele as Coventry left themselves exposed at the back but each time the keeper was solid.
Substitute Jennison Myrie-Williams had one long breakaway and, after deciding not to free a team-mate, shot from the edge of the box again forcing Steele into action.
Not for the first time this season Adams had to endure calls for his head from the main home stand and, most worryingly for the manager, there was precious little reaction from his side.