David Cotterill's quickly-taken free kick proved decisive, before both teams finished with 10 men as Rovers striker Iain Hume and Latics goalkeeper Dean Bouzanis were both sent off - Hume for a stoppage-time foul and Bouzanis for pushing him in the resulting fracas.
Both players will now miss out when the sides resume hostilities in Saturday's FA Cup second-round clash.
"I'm really pleased for all the supporters because they turned out in force on Saturday and gave us great backing and they've helped us win the game here," Saunders said.
"I think that's something like eight wins in the last 10 games and it's all credit to the players because they just keep churning out results.
"We had some injury problems and then lost a couple of players during the course of the game but they showed great character to overcome everything and take the points. We had to lift ourselves in the second half and raise our game because Oldham came and had a go and I thought they played well."