The frontman, on loan from Manchester United, came off the bench to score the winner after Franck Moussa's opener had been cancelled out by Liam Noble's second-half strike.
The result leaves the Cumbrians six points clear of the relegation zone, one place and one point below the Sky Blues in the League One table.
Following an uneventful opening half an hour it was the hosts who took the lead after 32 minutes.
Conor Thomas' excellent pass found Moussa unmarked at the back post and the Belgian delicately slotted home his 14th goal of the season.
But the visitors came out the stronger in the second half and found the equaliser after 55 minutes, an excellent back heel from Liam Millar finding Noble unmarked and he finished into the corner.
United then completed their comeback as Byrne latched on to John Fleck's poor header and finished emphatically past Joe Murphy after 70 minutes to secure the club's first away win since October.