Brown latched on to a Marcus Stewart flick and fired home a low shot late on to give them a sixth straight win.
He could have settled it earlier but was denied by the woodwork, while fellow replacement Sean Thornton and Julio Arca also spurned good chances.
Stern John, Trevor Benjamin and Gary McSheffrey went close for Coventry, who slipped into the bottom three.
Coventry boss Micky Adams:
"It's not a psychological blow at all. If we'd laid down, let Sunderland tickle our bellies and not taken part in the game, it would be.
"I didn't want to just sit back and let them beat us. I wanted us to be positive and to try and win the game.
"On chances created, especially in the first half, I think we did well."
Sunderland: Myhre, Wright, Breen, Caldwell, McCartney, Lawrence (Brown 55), Whitley (Thornton 55), Robinson, Arca (Welsh 68), Elliott, Stewart.
Subs Not Used: Ingham, Danny Collins.
Booked: Stewart.
Goals: Brown 76.
Coventry: Bennett, Duffy (Shaw 83), Staunton, Page, Hall, Jorgensen, Doyle, Osbourne, McSheffrey (Adebola 77), John, Benjamin (Barrett 83).
Subs Not Used: Steele, Morrell.
Booked: Osbourne.
Att: 29,424.
Ref: A Wiley (Staffordshire).