Burch produced a world-class save 20 minutes from time to tip away a top corner effort from Bradford winger Omar Daley.
The save was the best of a host of stops made by the young Lincoln keeper to deny the visitors in an entertaining game.
Referee Neil Swarbrick waived away early penalty appeals from Daley and Bradford team-mate Nicky Law, but it was Lincoln who looked the more dangerous in open play.
The home side created the first real chance when centre-half Danny Hone got on the end of an Aaron Brown corner, and saw his far-post header cleared off the line by Bradford striker Michael Boulding.
Lincoln winger Dany N'Guessan then fired wide with Bradford keeper Rhys Evans stranded in no-man's land.
Evans then held another fizzing shot from N'Guessan at the second attempt.
Bradford were temporarily reduced to ten men on 36 minutes when skipper Graeme Lee was forced off the field for treatment to a heal injury, but it seemed to stir the visitors, and by the time Lee returned in first-half injury time Bradford were finally beginning to trouble the home defence.
Boulding managed to lose his marker, but wasted Bradford's only real effort of the first half when he went for the top corner but blazed wide.
Lincoln again took control after half-time and Lee Frecklington should have done better when he was set-up by N'Guessan, but fired wide when unmarked 12 yards out.
The game continued to flow from end to end and Burch had to get down low to push away a goalbound header from Boulding.
At the other end, Frecklington again went close when a better effort was smashed away by Evans, diving full-length across his goal.
Bradford looked more dangerous in the closing quarter of the game and it was left to Burch to produce a memorable save to deny Daley's 25-yard shot.