Pittsburgh (6-7) led 31-6 when James Conner, the Atlantic Coast Conference offensive player of the year, got his second touchdown run with 14 minutes left in the game.
Houston (8-5) went on to the biggest comeback in an FBS game this season and matched the third-largest in a bowl game.
Ward had an eight-yard scoring pass to Deontay Greenberry before the Cougars recovered their first onside kick. That led to a 29-yard catch by Demarcus Ayers on a fourth-and-13 play with 1:58 left, and Houston tried another onside kick.
While the second one didn't go 10 yards on the rain-slickened field, the ball was touched by the same Pitt defender who couldn't handle the first one before Houston running back Kenneth Farrow jumped on it at the Cougars' 43-yard line. Greenberry started the drive with a 38-yard catch, then on third and 16 had a 25-yard touchdown with 59 seconds left.
Interim coach David Gibbs opted for a win-or-lose two-point conversion try, and Greenberry made a leaping catch in the back corner of the end zone.
The Volunteers (7-6) scored on their first four possessions, leading 28-0 before Iowa (7-6) managed 70 yards.
Hurd, Dobbs and a bit of trickery helped Tennessee build the big lead.
Hurd broke tackles on nearly every run, capping Tennessee's first possession with a three-yard score and adding a 29-yard touchdown scamper on the next drive.
Tennessee made it 21-0 late in the first quarter when Dobbs threw a lateral to running back Marlin Lane in the right flat. Lane, a senior from nearby Daytona Beach, turned and hit Vic Wharton in stride down the sideline.