Franck Moussa's 14th of the season opened the scoring for the Sky Blues in the first half, but two second-half strikes from Liam Noble and substitute Sam Byrne secured all three points for the Cumbrians and their first away win since October.
The victory also lifts United six points away from the relegation zone and boss Kavanagh was delighted with the character his team showed to come from behind.
"I am extremely proud and extremely pleased with the players," he said.
"On the whole, the performance, the energy and the work-rate was excellent and I'm really pleased.
"I have to pay big tribute to the players. I had a bit of a go at them at half-time because they respected Coventry too much.
"I asked them to step on 10 yards in the second half and we managed to get our full-backs on the ball and that was a real feature of our play.
"It was a very organised rigid defensive display and it was a really good all-round performance. Back-to-back victories now will give us confidence.
"You don't usually get a win coming away from home when you lose the first goal, but we have here and that is down to the players."
Coventry had the better of the first half and after Moussa's strike they should have doubled their lead, but debutant Chuba Akpom was denied by a marvellous save from Jordan Pickford.
Carlisle came out rejuvenated after the break and, after Noble's fine strike, substitute Byrne, on loan from Manchester United, secured the win after he latched on to John Fleck's poor header with 30 minutes to play.
- Sports & Recreation
- Franck Moussa
- Liam Noble