Jarvis, who switched between the clubs in the summer, fired home from close range after keeper Martin Rice had palmed out David McGurk's goal-bound header.
City striker Wes Fletcher scored a brilliant 53rd-minute opener on the spin, controlling the ball well with his back to goal before lashing it past unguarded Rice from close range.
The goal came on a counter-attack after City keeper Michael Ingham denied Billy Bodin, who also shot on the spin.
Seconds earlier the York keeper saved well to keep out a Karl Hawley angled shot after he was put in on goal by substitute Damon Lathrope.
Northern Ireland international attacking midfielder Josh Carson made it 2-0 with a thumping free-kick from 25 yards out into the top corner off the bar after Rice got his hands to the shot.
Torquay had the better chances in the first half, with striker Hawley heading just over from left back Kevin Nicholson's 13th minute cross.