After securing just five points in 2013, the Gunners were set for a second successive home draw until 12 minutes from time.
Arsenal remain sixth in the standings after the 1-0 win, but closed the gap on fourth-place Tottenham to a single point - much to Wenger's relief.
"I said already before the West Ham and Liverpool games, that we are in a position where we cannot drop points," the Gunners boss said.
"We dropped two points against Liverpool but we couldn't do it again today.
"What is more important is the quality of performances is there. The spirit is there.
"The target to be in the top four and to focus on the two cups we are still in is very important now as we have strengthened our belief and our confidence."