Montrose took the lead in the sixth minute when Adrian Mallagaray hit the back of the net from the centre of the penalty area.
The amateurs took the lead eight minutes later when goalkeeper Ross Salmon diverted the ball into his own net, and Ryan McGeever doubled Queen's lead three minutes later, finding the bottom-left corner from close range.
Vincent Berry came close to extending the lead on the hour mark but his long-range shot struck the right post.
Woods added a fourth with 10 minutes remaining when his shot found the bottom-left corner.
There was still time for a fifth goal, and Jamie McKernon's shot struck the top-left corner with Queen's next attack.
A sixth came in stoppage time when Burns's free kick dipped into the net to add gloss to the commanding victory.