Mesut Ozil scored the winner on 71 minutes at the Riverside Stadium after Alvaro Negredo had cancelled out Alexis Sanchez's 42nd minute opener, five minutes after the restart.
The three points lifted the Gunners onto 57 points, in sixth place, to just about keep their European hopes alive.
After the match, Wenger told reporters: "It was a committed game that we needed to win.
"I thought in the first half it was tight because they defended very well and we were tested just after half-time because it came back to 1-1 and then we were tested mentally and we found the response we needed and that’s what we wanted.
"It was one of the last chances to stay in the league and that’s why we never gave up. We were a bit maybe riding our luck a little bit in one or two occasions on crosses, but overall we were a bit more solid defensively today."