Dunn swept home past his former Rovers team-mate Brad Friedel after Richard Dunne handled in the box, earning the three points despite Vince Grella's 68th-minute red card for two bookable offences. Christopher Samba had put the hosts on level terms in the first half.
The manner of the win, with the home side clawing back from the shock of conceding a third minute opener to Gabriel Agbonlahor, convinced Allardyce his side has what it takes to climb the table.
"We've burst all the odds and it shows the determination of our squad and the quality of our team," said a visibly delighted Allardyce. "Now what we have to do is hit those consistent performances.
"I'm delighted with the way we kicked on after going a goal down in the first few minutes. It really only knocked us back for a few minutes. We came storming back and we got what we deserved."