Injury time was approaching when Broghammer connected with Eliot Richards' low right-wing cross to level the scores.
David Clarkson had put Rovers in front with a stunning 30-yard strike after 11 minutes, but seven minutes later Craig Reid equalised with a deft lob after controlling a long ball from Troy Brown.
Alex Rodman was allowed space just before the hour mark to lash an angled right-footed shot across Rovers goalkeeper Sam Walker and inside the far post from 12 yards to put the Shots in front.
Visiting goalkeeper Jamie Young then made fine saves from Matt Harrold and Broghammer to protect his side's lead until the late drama.
The substitute's equaliser saved Rovers from a fourth defeat in as many home games although they again looked short of ideas going forward.