Manager Mick McCarthy made a double substitution in the 56th minute, hauling off both Liam Lawrence and Jeff Whitley for Brown and Sean Thornton, after his side failed to make any impression on a Sky Blues outfit battling for survival at the other end of the table.
Brown struck 20 minutes later to open up a five-point gap over third-placed Ipswich in the chase for an automatic return to the Premiership.
City failed to seriously test home goalkeeper Thomas Myhre despite carving out several clear-cut chances and the defeat leaves Micky Adams' side still with it all to do to stave off relegation.
Coventry enjoyed the better of a lacklustre first half and should have made the Black Cats pay for a disjointed performance.
Stern John set the tone from the sixth minute when he headed just past the post from Michael Doyle's corner, Richard Duffy sent a volley inches past the post with Myhre stranded and Marcus Hall headed another corner off target.
Sunderland's response was Carl Robinson sending a low drive just past the post when set up by the lively Stephen Elliott before goalkeeper Ian Bennett saved with his legs from Julio Arca.
Coventry continued to press and Claus Jorgensen sent a volley over the bar before Duffy fired a fierce drive wide of the target.
Sunderland upped the tempo after the substitutions and Elliott forced Bennett into action before Brown squandered a great chance when sent clear in the 67th minute, crashing his low drive against the foot to the post.
The home side finally made the breakthrough in the 76th minute. Marcus Stewart sent Brown clear and the 20-year-old made amends for his earlier miss by confidently sliding the ball past Bennett.
Adams responded by throwing on three substitutes but, apart from John heading just over in added time, City failed to threaten Myrhe's goal.