The Black Cats had goalkeeper Jon McLaughlin to thank for their clean sheet after he pulled off a string of excellent saves to preserve the lead his side gained in the home leg at the Stadium Of Light last weekend.
McLaughlin began his heroics with a point-blank stop from Gareth Evans in the first half, who charged in to head a knocked-down cross only to be denied by the sprawling keeper.
The Scottish stopper was equally impressive in the second period, denying Oliver Hawkins and substitute James Vaughan, who also put excellent headers on target.
Sunderland will play either Charlton Athletic or Doncaster Rovers for a place in the EFL Championship next season.
That tie is in Charlton's favour after it won the first leg 3-2 away from home.