The fiery midfielder had replaced Leon Britton five minutes earlier when he struck a left-foot shot into the back of the net after a goalmouth melee.
The Swans had Adebayo Akinfenwa leading the line in place of record signing Rory Fallon and Dennis Lawrence was back at the heart of the defence after recovering from a groin strain.
Thomas Butler returned to a wide midfield role while Robinson had to be content with a place on the bench.
The Swans won a couple of corners in the opening minutes as Akinfenwa made his bulky presence felt and, with Lee Trundle working his way through the visitors rearguard at will, it was constant Swansea pressure during the opening quarter.
Despite the dominance of the hosts it was Bradford who went closest to an early goal when Jermaine Johnson struck a volley from out wide on the right and the ball flashed over the Swansea crossbar.
Six minutes later veteran striker Dean Windass was cleverly worked through but Willy Gueret came out swiftly to block.
On the half-hour Butler rifled a shot against the foot of the post with Donovan Ricketts rooted to the spot.
Akinfenwa looked to have scored in the 40th minute but, as his header went towards an empty net, Bantams skipper David Wetherall raced back to clear.
Bradford suffered a blow on the stroke of half-time when the influential Windass left the field with a foot injury.
Yet another chance went begging 10 minutes after the restart when a cross from the busy Trundle put Akinfenwa in space, but once again he failed to beat the visitors' keeper from close range.
When Robinson took the field the change had the desired effect and in a goalmouth scramble he hooked in the winner.
Suddenly the hosts were playing with more freedom and Trundle burst through only to curl his shot against the post.
The Bantams pressed hard to level but the home defence held firm for a deserved victory.