Darren Fletcher's side, watched by Michael Carrick, his coaching staff and former Reds Wayne Rooney and John O'Shea, were in good form at Carrington and pulled clear of the spirited visitors.
Cameron Byrne-Hughes also saved a penalty from Rams striker Israel Green in the second half, as United close the gap on leaders Manchester City to a point.
FIRST HALF - REDS GAIN CONTROL
United started with purpose, Bradbury shooting wide early on, and were in front come the sixth minute. Derby keeper Cristiano Dzialuk only got a touch to a corner that was sent across goal by Jay McEvoy, at full stretch, and Bradbury was on hand to tap in his fifth goal of the season for the Under-18s.