Bradshaw, who has missed 19 games with hamstring problems during the season, opened the scoring in the 18th minute when he chipped goalkeeper Joel Coleman from Jordy Hiwula’s pass.
And he added the second six minutes later from the penalty spot after Malvind Benning was brought down by Latics’ skipper Liam Kelly.
“It’s a great achievement to score 29 goals at any level, and I am very pleased for him,” he said.
“I have said in the past he works very hard and is a lovely lad. The type we like to have around the place.
“I’m not sure it was a penalty, but it’s the first we have had here this season. It was not as clear as the one we didn’t get against Port Vale recently. I didn’t think we were at our best in the second half, but they (Oldham) have some good players and Dean (Holden, the Latics’ manager) got them working hard.”
Walsall end the season away at Bristol City, the champions who beat them in the Johnstone’s Paint Trophy final at Wembley, and Smith said: “We want to finish the season as high as we can and be unbeaten in our last eight games.”
Oldham improved considerably in the second half but couldn’t put the finishing touch to some good moves, and manager Dean Holden said: “The fact is the second goal was not a penalty, but I am not blaming the referee for the defeat.
“We have been missing our top scorer for weeks, and we were not positive enough in the final third. I was disappointed for the fans, some in fancy dress, because they got behind us. We had a go in the second half, and I didn’t want the game to become three, four or five.
“Decisions have gone against us, but we have to overcome adversity.
“It was a disappointing, flat performance, certainly in the final third. I have asked the players how many thought they six or seven from that performance. We were not positive enough.”
After a poor first half Oldham were much improved after half-time, but couldn’t penetrate a Walsall defence who have a good record for keeping clean sheets this season.
Hard working Mike Jones hammered on shot over the bar, but Hiwula, on loan from Manchester City, missed two good chances to add to Walsall’s tally in the second half.
Walsall boss Smith again insisted he knew nothing about speculation that Millwall would like him as their next manager. He has another year to go on his contract with the Saddlers.