The long trip back to the South Coast looked like being a disappointing one for the visitors after Danny Cadamarteri's scrappy opener had put City in front, but a super save by Gareth Stewart kept them in the hunt and they ran out comfortable winners in the end.
Sean O'Driscoll's men had restricted free-scoring City to a handful of half-chances and would have been well pleased to go in level at the break, but in injury time disaster struck with the interval just seconds away.
Skipper Marcus Browning was penalised for bundling over City hot-shot Dean Windass and the seven-goal striker stepped up to rattle the crossbar with his well-struck free-kick.
But the Bournemouth defence reacted slowly as the loose ball came down invitingly in the six-yard box and Cadamarteri was quick to claim the last touch in a goal-line scramble.
Cherries' dangerman James Hayter had come closest to hitting the Bantams net in the first period with his fierce left-foot drive, but a game of few openings could have been put beyond them had Windass struck just after the break.
After 48 minutes Cadamarteri robbed Brian Stock in midfield and Stewart, making his first league start after a series of injuries, did superbly to push away Windass' rasping drive from the edge of the box.
The Cherries responded well to the let-off but it took a large slice of luck to get them level.
Hayter, who proved a constant threat, looked to be going nowhere down the right wing, but when he whipped over a teasing cross, City centre-half Mark Bower flung himself at the ball and headed past stranded keeper Russell Howarth.
With Stock and Browning having more and more influence, the visitors sensed a first win at Valley Parade for 10 years and Surman provided the perfect strike.
The midfielder was given far too much room on the edge of the Bradford box, but there was no dispute about his right-foot curler which beat Howarth all-ends-up.
City were lucky to escape a penalty shout after Surman was sent crashing in the box by Ben Muirhead, but keeper Stewart was rarely tested in the final quarter.
Bradford substitute Andy Cooke came closest to equalising when his excellent left-foot volley clipped the crossbar but the home side were left to rue the victory that slipped through their fingers.