Tomas Rosicky's double had Arsenal in control, but West Brom pulled a goal back through James Morrison's penalty after Per Mertesacker had been sent off for bringing down Shane Long.
However, the visitors held on grimly and, whilst Wenger admitted previous Arsenal sides may have conceded, he hailed the spirit and resilience of his 10 men.
"It was a fabulous win and it needed a special performance," Wenger said. "The last 20 minutes was obviously very nervy because we had 10 men.
"They had a lot of strong bodies who were very determined to score. But we kept our nerve and we were committed.
"We have good spirit, the team is growing and we have a resolute attitude. And, of course, the run we are on makes the team stronger."
The result moved Arsenal into the top four for the first time since Boxing Day, with Tottenham, Chelsea and Everton playing on Sunday, and Wenger believes his team can stay in the UEFA Champions League places.
He added: "I am very pleased, it is three big points. We have 22 points out of the last 27 but we know our target is to continue to do that until the end of the season.
"We know we have one target to win our games, then we don't have to look at Tottenham or Chelsea. If we lose our games, then you have to look at the results of the others. Our attitude is simple, we have to win the games.
"Can we finish fourth? I believe we will do it. We are conscious as well we have to give 100 per cent commitment in every game and it will be difficult and edgy to the end.
"We are ready for the fight and we always tell the players 'don't look at everyone else, just focus on ourselves'."