An outswinging corner from Martin Grainger was flicked on by Tommy Mooney, and Purse slammed the ball home from close range.
Bradford responded, with Lee Sharpe forcing Nico Vaesen into a diving stop after nine minutes.
Paul Devlin came close to adding a second for Blues in the 12th minute.
When Stern John was brought down on the edge of the area, Devlin floated the free-kick over the wall, but straight into the arms of Bradford keeper Alan Combe.
Jess almost equalised in the 26th minute. Sharpe's corner came out to him, but his effort was well held by Vaesen.
Bradford were made to pay for this missed chance as Blues extended their lead in the 31st minute.
The Blues almost made the game safe just two minutes later, but Devlin's shot was caught by Combe.
Bradford lost midfielder Gareth Whalley with an injury in the 44th minute - he was replaced by Juanjo - and just a minute after his introduction, the Bantams were back in the game.
Stuart McCall cushioned a header into the path of Jess, and he struck the ball into the top corner of the net for his 13th goal of the season.
The second half started quietly, with Cadamarteri's 54th-minute shot the first real opening.
Blues replaced new signing Damien Johnson with Stan Lazaridis in the 58th minute, as Bradford began to look more threatening.
Horsfield came on to replace Mooney in the 69th minute, and 12 minutes later he sealed the points for Blues on the counter-attack.
A short backpass by Bradford defender Jacobs was intercepted by The Horse and he slid the ball home to secure victory.
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