The hard-earned victory moves City up into second place in Under-18 Premier League North on 21 points just two behind leaders Blackburn.
Rovers, however, have a game in hand.
After a tight, cagey first 45 minutes, Breckin struck to give City a 46th minute lead before substitute Dickson then extended the advantage on 83 minutes.
The Clarets pulled a goal back a minute from time, but City hung on to claim three vital points.
Watch the highlights on the video above.