The Reds led at the interval through Byrne's early opener and ran away with the game in the second half to clinch a place in the elite phase of the Barclays Under-18 Premier League.
After smashing North-East rivals Newcastle 6-1 in their last outing, the Black Cats must have had high hopes of upsetting Paul McGuinness' in-form side but fell behind on nine minutes when Irish striker Byrne, who had already missed a good chance, tapped in from close range after Barmby won possession high up the pitch.
Callum Hope hit a post for Sunderland on 25 minutes and that was as good as it got for the home side with Byrne almost adding a second before the break when his effort was well saved by Jonny Maddison.
Mikael Mandron spurned the hosts' clearest opening when heading over the bar from six yards out on 70 minutes and substitute Wilson put the game beyond reach with a quickfire double. The Reds' top scorer punished some sloppy defending to claim his ninth goal of the campaign and reached double figures four minutes later.
Incredibly, Wilson now has seven goals in the North East this term as he turned home following a well-worked short-corner that allowed Mats Daehli to get away down the byline and cut back for the prolific marksman.
Barmby added a fourth with a fine, curling effort that evoked memories of Ashley Young's first strike in the 8-2 win against Arsenal last season, swirling a beautiful shot past Maddison and into the top corner of the net.