Everton were the better side throughout and took the lead with Sylvain Distin’s back post header before half time.
Both sides were denied penalties, as the Black Cats were unable to find a way through for an equaliser.
The win sees Everton finish second in Group D ahead of Villarreal, and sets up a clash in the next round away to Group C victors Leicester City.
It took a while for the game to get going, with soon-to-be goalscorer Distin having the game’s first scoring opportunity. But his tame header from a corner was easily saved by Sunderland goalkeeper Joel Dixon.
The Frenchman’s fellow first teamer Leon Osman was the next player to come close to breaking the deadlock, but he couldn’t get enough behind his curling shot to thwart the Sunderland shot stopper.
And the Black Cats offered an almost immediate reply of their own, as Thomas Beadling’s glancing header went agonisingly close to opening the scoring.
It was a wonder the game remained goalless through 15 minutes after a moment of madness from Dixon.
The goalkeeper rushed out of his goal to stop the oncoming David Henen, but the Belgian managed to take the ball past his adversary before Dixon recovered fantastically to gather the winger’s centre.
There was more defensive panic for the Black Cats just minutes later, when the ball appeared to strike centre back Liam Agnew on the arm; but referee Simeon Lucas quickly waved away stern Everton protests.
Henen was among the action again on the half hour mark, when he latched on to an Osman through ball to go through on goal, though he couldn’t steer his shot on target, and Sunderland were left to breathe a sigh of relief.
But finally, David Unsworth’s youngsters got the breakthrough – thanks to an older member of the squad.
Henen was brought down on the edge of the area, and George Green took the resulting free kick finding 37-year-old Distin at the back post who planted the ball into the back of the net with a neat header.
The Toffees had another unsuccessful shout for a penalty shortly after the restart when Distin, again embroiled in the action, appeared to be dragged to the turf by Beadling in the area.
The second half went by without much incident, though Gethin Jones should have put the game beyond doubt ten minutes from time, but he was unable to beat the impressive Dixon who made the save.
The quarter final games are as follows:
Chelsea U21 vs Fulham U21
Athletic Bilbao U21 vs Porto U21
Leicester City U21 vs Everton U21
Celtic U21 vs Manchester City U21
Everton U21: Griffiths (GK), Kenny, Galloway, Junior, Distin, Browning, Jones, Lundstram (Ledson, 46’), Henen (Donohue, 83’), Osman, Green (Connolly, 72’) Subs: Hunt, Ledson, Dowell, Walsh, Langton, Connolly, Donohue
Scorers: Distin (38’)
Sunderland U21: Dixon (GK), Cartwright (Greenwood, 71’), T Robson (Ledger, 71’), Agnew, McNamee, Beadling, Watmore, Smith, Mandron, Honeyman (Blinco, 79'), Lawson Subs: Greenwood, Talbot, Wright, J Robson, Nelson, Blinco, Ledger
Referee: Simeon Lucas (Lancs)