The Swansea kepeer made two excellent saves to deny Steve Claridge as City poured forward in the frantic closing stages.
The visitors had started the better of the two sides and City's own keeper Donovan Ricketts had to be at his best to keep out Leon Knight's swerving shot after 25 minutes and then produced a sensational stop to deny striker Adebayo Akinfenwa.
His opposite number, Gueret, had little to do at the other end as City struggled to break down the Swansea defence.
Todd had moved Dean Windass into midfield and gave teenage striker Joe Brown his first league start alongside 39-year-old Claridge in attack as he looked for a cure to his side's poor home form.
But the changes had little effect in the opening period and Swansea took the lead just a minute before the break.
City thought they had cleared the danger from a corner but Roberto Martinez found Tom Williams in space on the right hand side. His low centre found Gary Monk unmarked and he sided-footed the ball into the roof of the net.
Bradford though came out fighting at the start of the second half and nearly scrambled themselves and equaliser but Gueret did well to turn behind Claridge's effort.
With 30 minutes to go Todd brought on Tom Penford for his seasonal debut in place of the ineffectual Bobby Petta and the youngster showed some excellent touches before setting up Windass for the equaliser.
He floated in a tantalising cross which City's leading scorer buried with a bullet header.
As the crowd got behind them, City went looking for a winner but were denied two amazing injury time saves from Gueret.
First he flung himself too his left to tip wide Claridge's goal-bound header and then acrobatically tipped over a fine volley from the same man and from the resulting corner Mark Bower fired over.
Despite being disappointed not to have claimed all three points in the frantic finale, Todd will take great encouragement by the way his side fought back against a strong Swansea outfit.