The 37-year-old striker rounded off a superb three-man move, hooking the ball into the corner of the net from eight yards after left-back Craig Woodman's cross to the far side of the penalty area had been headed into the goalmouth by Matt Oakley.
Cureton had come close to scoring in the 13th minute when he saw his 15-yard shot rebound off the far post after Alan Gow had played the ball into his path.
The Exeter defence withstood some intense pressure in the second half as Bradford pressed for an equaliser, while the home side had a strong claim for a penalty in the 58th minute when leading scorer Nahki Wells appeared to be pulled by right-back Steve Tully.
Skipper Danny Coles cleared off the line after substitute Alan Connell forced the ball past 'keeper Artur Krysiak in the 73rd minute.
Fellow Bantams substitute Garry Thompson missed a good chance, shooting wide from 10 yards six minutes from time, as the visitors survived to condemn Bradford to only their second home defeat of the season.