Sam Baldock's penalty in first-half stoppage time was enough to secure the Robins their fifth victory in nine games - a sequence that has seen the club reach 50 points and move seven points clear of the League One drop zone.
"We are probably safe now, but only probably," Cotterill told reporters. "We have four games left to play, after the start of the season was a bit of a disaster, and we now want to get ourselves into, maybe, the top 10.
"Walsall are a good team and I said that when they played at our place, and a penalty separated us that day.
"Walsall are a really good team and Dean [Smith] has done a fantastic job here, and they have been up there all season. I knew it would be a difficult game."
City have now won their last four games against Walsall without conceding a goal.