Torquay, who had skipper Brian Saah sent off for elbowing James Meredith in the 35th minute, denied Bradford with some desperate defending before Connell broke the deadlock with a superb right-foot volley from the edge of the penalty area from Nahki Wells' cross five minutes after coming on as substitute.
City ought to have been ahead after dominating the first half. Striker James Hanson's header was turned over the crossbar by goalkeeper Michael Poke in the second minute and Hanson then saw another header rebound off the bar two minutes before half-time.
Torquay also hit the woodwork in the 39th minute when substitute Niall Thompson turned sharply only to see his 20-yard shot hit a post.
Poke continued to deny Bradford in the second half, turning Wells' goalbound shot on to a post in the 70th minute. But he had no chance with Connell's winner.