On-loan midfielder Tom Adeyemi gave Bradford a 66th-minute lead, firing high into the net from 25 yards after visiting goalkeeper Ben Smith had punched Luke O'Brien's free-kick into his path before Bradshaw entered the fray.
Bradshaw, one of two substitutions made by Shrewsbury manager Graham Turner in the 75th minute, equalised with an angled shot two minutes later when David Raven's throw-in was flicked into his path before scoring the winner three minutes from the end.
Jon Taylor created the chance, going past two Bradford players down the left-hand touchline before cutting in and laying the ball into the path of Bradshaw, who scored the winner from close range.
Source: PA
Source: PA