The Gulls had to play with a man less for 21 minutes after Krystian Pearce was given his marching orders.
In a scrappy first half of few chances, recalled Wales Under 21s winger Billy Bodin came closest to breaking the deadlock on 20 minutes.
Bodin hit the inside of the post with a thumping left-foot shot from 20 yards that beat diving Stags goalkeeper Alan Marriott.
The visiting stopper made a good save at the start of the second half, denying United's on-loan Charlton teenager Adebayo Azeez from six yards out.
Mansfield countered with intent and Torquay keeper Martin Rice made a superb save on the line after striker Matt Rhead had sent Jamie McGuire's cross towards goal with a clever stooping header.
Left-back Kevin Nicholson drilled a 30-yard free-kick just over in the 68th minute but a minute later Torquay were reduced to 10 men as central defender Pearce was sent off for a second caution after hauling back Sam Clucas.
However, the Stags could not find a break through despite having the numerical advantage.