The Wales youth international was right among the thick of the action at the Clarets' Training Centre, and forced several good saves out of Burnley goalkeeper Callum Feeney, especially in a first half that United dominated.
But Feeney couldn't prevent Bradbury from opening the scoring just before the interval, when connecting with Nathaniel Junior Brown's exquisite cross.
There looked to be a sting in the tail when, after more excellent Feeney stops, Luca Scanlon - former Red and brother of United Academy winger James - nodded the hosts level after 79 minutes.
However, Bradbury was in the right place at the right time in the dying seconds, as his composed finish gave Darren Fletcher's side the three points required to leapfrog Nottingham Forest into second place, on 29 points, in the U18s Premier League North table.