The former Sheffield United and Scarborough striker scored either side of half time as the Shakers ended a run of three successive defeats.
Adam Lockwood headed Torquay into a ninth minute lead and, it was against the run of play, when Bury equalised in fortuitous circumstances. Matt Hewlett will be credited with an own goal, though Dave Flitcroft will try and claim it for a shot that deflected off Hewlett over Andy Marriott.
There was no dispute about the second as Speight headed in from six yards following Flitcroft's cross.
And Torquay's collapse completed when Speight added a second - all three Bury goals coming in a 10-minute spell.
The visitors, who didn't concede against Premiership Birmingham in the FA Cup to earn a replay on Tuesday, fought back in the latter stages.
Despite going close several times through substitute Morike Sako all they could muster was a stoppage time consolation from Jo Kuffour.