Danny Philliskirk was the only Oldham player to trouble keeper Craig MacGillivray but his low effort was easily saved.
Walsall, who extended their unbeaten run to seven games with a third straight home win, move up to 11th place in League One.
Against an Oldham side under the temporary management of former Saddlers player-coach Dean Holden, the Saddlers gave a debut in goal to MacGillivray, as they await a decision over first-choice Richard O’Donnell’s future.
Walsall manager Dean Smith told BBC WM:
“There were plenty of positives and seeing Tom Bradshaw reach 20 goals was certainly one of them. It’s a fantastic achievement for the lad.
“Our performance levels have been high in recent weeks. The players have deserved to keep their place and that’s why I haven’t made many changes.
“I was delighted that Craig MacGillivray came in and kept a clean sheet. He looked steady and calm on the ball and it was an accomplished senior debut from him.
“Whilst we are hopeful that Richard O’Donnell will stay, I had to take a look at Craig today and Richard understands that.”
Oldham boss Dean Holden told BBC Radio Manchester:
