The result extended the Saddlers’ unbeaten run to seven games, and despite an improved second half the Latics never looked like saving the game.
The hosts pushed forward from the start and took the lead in the 18th minute when Kieron Morris slid a neat pass through for Bradshaw to beat the offside trap and chip goalkeeper Joel Coleman as he rushed off his line.
Six minutes later Malvind Benning was hauled down by Oldham skipper Liam Kelly as he burst into the box and Bradshaw hammered the resulting penalty home.
The visitors offered little going forward and home keeper Craig MacGillvray, making his debut for the club, had only one save to make, smothering a low shot form Danny Philliskirk on the half hour mark.
Oldham had more of the play in the second half, but Jordy Hiwula missed two good chances to extend the lead for Walsall.