Bradshaw gave the Saddlers the lead on 18 minutes and he doubled their lead from the penalty spot on 24 minutes to seal the win for the Saddlers.
Walsall dominated the opening proceedings at the Banks's Stadium and could have taken the lead on five minutes.
Adam Chambers fed the ball into the path of Bradshaw but his shot was tipped away by goalkeeper, Joel Coleman.
Latics, who were backed by a loud and colourful set of supporters, were struggling to get into the game with Rhys Turner and Danny Philliskirk struggling to get hold of the ball.
Walsall were threatening on the break and they took the lead on 18 minutes through Bradshaw.
Athletic had a corner but Walsall managed to clear the danger and Bradshaw raced away to knock the ball beyond Coleman to give the Saddlers the lead.
One became two as Walsall extended their lead from the penalty spot.
Liam Kelly looked to win the ball fairly inside the penalty box from Malvind Benning but the referee pointed to the spot and Bradshaw sent Coleman the wrong way to double their advantage.
The referee was proving to be very unpopular with the visiting supporters as made a couple of decisions which didn't favour Latics.
Philliskirk had Latics' first shot of the first half on the half hour mark but his snapshot from the edge of the box was comfortably saved by Craig MacGillivray.
Latics' players felt they should have been awarded a penalty five minutes before the break as the ball appeared to hit the arm of Ben Purkiss but the referee waved away the protests.
Latics made a change at the break as Carl Winchester replaced Timothee Dieng and Latics lined up in a more attacking formation.
Athletic started the second half with a lot more urgency and started to put pressure on the Walsall defence for the first time in the game.
Mike Jones, Turner and Dominic Poleon were all putting themselves around and pressurising the Sadlers rearguard.
Latics had their first real chance of the game on 55 minutes as Kelly's inswinging cross almost found the head of the onrushing Philliskirk but the Walsall goalkeeper was quickly out to deny the forward.
It was all Latics but again, they had to be wary of their opposition on the counter attack.
On the hour mark, Walsall should have put the game beyond doubt but Kelly's last ditch block on Hiwula prevented him from making it three.
Jordan Bove and Rhys Murphy took to the field in place of Poleon and Turner as Dean Holden looked to get something out of the game.
Although Latics were enjoying a lot of the possession, they were struggling to create clear cut chances to test MacGillivray in the Walsall goal.
The home side could have made it three in the final minute of the game but Coleman stood up well to deny Manchester loanee Jordy Hiwula from finishing.
Latics forced a late corner but it wasn't enough to get anything out of the game and they travel back to Lancashire pointless.
Walsall: MacGillivary, Purkiss, O'Connor, A.Chambers (Flanagan 46'), Mantom, Bradshaw (Cook 84'), Cain (Sawyers 79'), Benning, J.Chambers, Morris, Hiwula.
Subs not used: Taylor, Downing, Forde, O'Donnell.
Oldham Athletic: Coleman, Brown, J.Wilson, Lockwood, Mellor, Dieng (Winchester 46'), Kelly, Jones, Philliskirk, Poleon (Bove 61'), Turner (Murphy .
Subs not used: Elokobi, B.Wilson, Truelove, Kean.
Goals: Walsall (Bradshaw 18', 24') 2-0 Oldham Athletic