Smith pounced on a defensive mix-up between Bradford full-back Lewis Emanuel and goalkeeper Donovan Ricketts to nip in and poke Paul Hall's hopeful long punt in off the post.
The result meant that Chesterfield avoided setting an unwanted football league record of nine successive draws.
But it was harsh on Bradford keeper Ricketts who had made good saves from Colin Larkin and Wayne Allison to keep the scores level.
Earlier Bradford made a bright start and had a Dean Windass goal disallowed for offside on 10 minutes.
David Wetherall and Bobby Petta also came close in the first half as Colin Todd's men looked the more likely to open the deadlock.
But the hosts stepped up after the break and Ricketts had to be at his best to keep out Hall's overhead kick and point-blank header from Allison.
However, with just two minutes to go Ricketts and Emanuel failed to communicate properly and Smith stole in to roll the ball home from just inside the penalty area.