Jordan Beyer scored his first Burnley goal nine minutes from time to open up a 16-point gap in the race for automatic promotion.
The German centre back headed home Josh Brownhill's corner just when it seemed the Clarets' winning run would end at six matches.
Coventry had dominated the first half, in striker Viktor Gyokeres had the outstanding player on the pitch, and defended doggedly as Burnley stepped up the pace with a welter of second half substitutions.
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