Dele Adebola, Christian Negouai and Gary McSheffrey all went close as the home side created a string of chances.
But Oster struck for Burnley on 65 minutes with a chip over keeper Luke Steele after Tony Grant set him up.
Coventry's fate was sealed four minutes later when Steele failed to hold on to a cross and Moore helped Grant's shot on its way into the net.
Burnley manager Steve Cotterill:
"We'd had to deal with plenty of pressure in the first half when we had to make sure we stood up to be counted instead of buckling.
"We did that and then I thought we were much the better side in the second half.
"We didn't go 4-4-2 out of any kind of disrespect for Coventry, it was just that we thought it would work for us today. I think we got it spot on tactically."
Coventry: Steele, Whing, Williams, Leacock, Staunton (Giddings 66), Barrett, Doyle, Hughes (Wood 45), McSheffrey, Negouai (John 46), Adebola.
Subs Not Used: Jorgensen, Shaw.
Burnley: Jensen, Duff, McGreal, Cahill, Camara, Oster (Roche 88), Hyde, Grant, Valois, Branch, Moore.
Subs Not Used: Coyne, O'Neill, Pilkington, Scott.
Booked: Hyde, McGreal, Duff.
Goals: Oster 64, Moore 69.
Ref: S Tanner (S Gloucestershire).