Garreth O'Connor rifled in a shot from the edge of the box to open the scoring and Wayne Thomas' near-post header doubled Burnley's 10 minutes later.
Steven Hughes was dismissed for foul and abusive language three minutes before the break to make things even more difficult for the Sky Blues.
Coventry boss Micky Adams:
"I can't defend Stephen. He's let himself down and his team-mates down.
"He's an experienced professional and I'll be disciplining him.
"It's bad enough being 2-0 down with 11 men, but you still have half a chance, but with 10 even that's gone."
Burnley: Jensen, Duff, Thomas, McGreal, Branch, Elliott (McCann 12), Hyde, James O'Connor, Garreth O'Connor, Akinbiyi, Noel-Williams, McCann (Lafferty 78).
Subs Not Used: Coyne, Sinclair, Yates.
Goals: Garreth O'Connor 28, Thomas 37, Akinbiyi 87, 90.
Coventry: Bywater, Duffy, Page, Heath, Hall, Scowcroft, Doyle, Hughes, McSheffrey (Williams 70), John (Jorgensen 45), Adebola (Morrell 76).
Subs Not Used: Ince, Wood.
Sent Off: Hughes (42).
Ref: N Miller (Durham).