Danny Whitaker put the home side ahead but goals from David Syers and James Hanson turned the game around before Bowery had the last word.
Chesterfield began the game with the confidence of a team flying high at the top of League Two and were in front in the 10th minute.
A slip let in Jack Lester and when the ball broke off the striker, Whitaker smashed in his 12th of the season from 10 yards.
But Bradford replied two minutes later when Gareth Evans crossed from the left and Syers headed in his eighth of the season at the back post.
The visitors took control and scored again in the 53rd minute when another cross by Evans was headed in by the unmarked James Hanson from eight yards. Bradford defended their lead skilfully but two minutes into added time, a header from Ian Breckin set up Bowery who fired under Jon McLaughlin.