The Gunners needed two spotkicks to overcome the Baggies.
Wenger said: "I am proud of the victory because it is important for our season that when your backs are to the wall, the team responds. Also because our spirit has been questioned because we had a flat performance last week. You can only come back and win on the football pitch.
"Somewhere, with top-level sportsmen, when you have to win, you win. To deal with the nerves and be serious and resolute, that is what we have shown."