The home side came close to going ahead early on when Wesley Burns' fierce effort whistled just over the bar.
Bradshaw latched on to Sam Mantom's smart backheel with a powerful strike to open the scoring from a tight angle just after the break.
Devante Cole forced Walsall keeper Neil Etheridge into a fantastic save deep into injury-time to preserve the win.
Walsall interim head coach Jon Whitney told BBC WM:
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"I've said to the lads, whenever I've been involved in promotion as a player and as staff, there's games that you remember.
"We knew this was going to be a tough, tough game. I thought they were unlucky at Burton.
"Credit to our lads, we dug in, we made the change when it was needed and I think after the change they struggled to live with us a little bit.
"Again our lads have come out on top with a fantastic 1-0 away win."
Walsall moved into third in League One after Tom Bradshaw's 17th goal of the season was enough to beat Fleetwood.