The 25-year-old, in the final year of his contract with relegated Barnsley, helped ensure it was back-to-back wins for the Reds - his seventh-minute strike being followed by substitute Victor Adeboyejo scoring with his first touch in the 70th-minute.
Barnsley, backed by a 2,552 following in the 18,986 crowd, were unchanged after the opening day's 4-0 win over Oxford.
The Bantams made just one change to the side which had won 1-0 at Shrewsbury with Eoin Doyle leading the attack in place of George Miller, who was on the bench.
Roadworks delayed the arrival of the Reds but they soon went into top gear after early City pressure.
Richard O'Donnell made a diving save to punch out a volley from Kieffer Moore, who had been denied by Anthony O'Connor's clearing header. The ball went wide and Dimitri Cavare whipped the ball back in from the right flank and Bradshaw's diving header beat the City goalkeeper in front of the Kop in the seventh minute.
Sean Scannell was inches away from converting a cross from Hope Akpan as City sought an immediate equaliser.
Barnsley's height up front, however, troubled the City rearguard and they could easily have fallen further behind.
City showed their intent when Doyle fired over from the edge of the area after a clever flick from Scannell.
Adam Davies also had to make a smart stop when Cavare could only volley a cross from Doyle at his goalkeeper.
Mamdou Thiam bent a shot wide for the Reds as half-time approached.
Sherwin Seedorf, nephew of former Dutch great Clarence, replaced Joe Riley for the Bantams at the break but Barnsley immediately went close, O'Donnell tipping over a drive from Alex Mowatt.
Ethan Pinnock's challenge on Scannell brough Bradford a 65th-minute corner rather than a penalty the Kop was baying for and O'Connor headed over from the flag kick.
George Moncur and Adeboyejo replaced Thiam and Moore in the 68th minute for the visitors.
The move worked wonders as Bradshaw launched the ball forward and Moncur held off Nat Knight-Percival as he moved the left before cutting inside and forcing the ball square past O'Donnell for a simple 70th-minute tap-in for Adeboyejo.
Tyrell Robinson and Miller were City's other changes, replacing Knight-Percival and Scannell.
It proved to no avail as the old refrain of 'It's just like watching Brazil,' rang out from the Barnsley fans before O’Donnell made a spectacular save from Bradshaw's header from a corner from Mowatt.
Six minutes of stoppage time failed to produce another goal as the Reds laid down a marker for the rest of the campaign.