Queens struck early when Ross Fergusson fed Dobbie, who unleashed a spectacular long-range shot that soared high into the net.
The visitors were two down as deadly Dobbie pounced again, teasing a couple of Brechin defenders before firing home. Dobbie was kept waiting for his treble on 25 minutes when Graeme Smith produced a fine diving save.
Dobbie was to fore again on the restart with a shot that Smith saved well, a stop that was matched at the other end when Alan Martin dived full length to turn a Callum Morrison shot behind.
But Dobbie was not to be denied and midway through the second half he produced another clinical finish to wrap up victory.
Brechin refused to buckle and got a consolation when Morrison linked with Andy Jackson before shooting low into the corner.