Angry Swindon fans stayed behind to chant for the sacking of manager Andy King who has presided over the miserable sequence of set backs.
Bradford trailed to an early goal but struck back to win more easily than the scoreline would suggest.
Swindon started brightly enough and took the lead with a superb goal from leading scorer Rory Fallon.
Left wing-back Andy Nicholas centered from the left and Fallon rose above the Bradford defence to head powerfully into the corner of the net.
Bradford had to wait until the 34th minute before getting level and even then there was a touch of good fortune about the equaliser.
Lee Crooks attempted a shot from outside the area but it was intercepted by Dean Windass who had time to turn and shoot past Tom Heaton from 10 yards.
Danny Cadamarteri should have scored again in the 40th minute when he raced clear on to a pass from Windass but shot straight at the on-rushing Heaton.
Claridge replaced Cadamarteri at the start of the second half and formed an attacking spearhead alongside Windass with a combined age of 75.
Nicholas headed wide when unmarked and Michael Pook went close with a shot from 20 yards but Claridge soon made himself the star man with two goals in eight minutes as Bradford took control.
Bobby Petta set up the first in the 64th minute by getting down the left and crossing low into the six-yard box for Claridge to shoot into the roof of the net from three yards.
Claridge's second was all his own work, collecting a pass from Windass, cutting in from the left on to his right foot and beating Heaton with a low shot.
Swindon pulled a goal back in the 76th minute when Fallon, who had been outstanding, hooked in a left-foot shot but it was too late to make any difference to the result.