The Reds overcame tough opponents from Slovakia after creating a number of chances but finding Milan Vincler in the home goal in good form. Jack Rudge and James Weir tested the keeper but he was exposed when Byrne slipped the ball into an empty net for the opener.
Mats Daehli came close while Adnan Januzaj again brought the best out of Vincler at the start of the second half. However, Tatran began to pose problems of their own with Matthew Wilkinson forced to clear off the line before an equaliser arrived on 70 minutes. Captain Filip Schlosser bundled home following a corner to make it 1-1.
United's response was swift as substitute Wilson profited from good play on the left-hand side to fire in the winner. Schlosser made a vital clearance to keep his side in the game as the Reds posted a victory against a fellow Under-18 team in front of a sizeable crowd.
Paul McGuinness' side are next in action on Wednesday against Dukla Banska Bystrica.
United: Pereira; Grimshaw (capt), McNair (Wilkinson 60), Dalley (McConnell 79), Love; Pearson (Goss 46), Rudge (Willock 79), Daehli (Harrop 79), Januzaj, Weir (Gorre 60); Byrne (Wilson 46).