The Czech Republic international has frequently struggled with injuries since arriving in north London from Borussia Dortmund in 2006 but has often impressed when fully fit.
Rosicky was making only his fifth Premier League appearance of the season at the Hawthorns but his return to form is particularly timely given the absence of the injured Jack Wilshere.
"He's had problems with injuries and then there was a problem with competition for places as well because you have Wilshere, you have [Santi] Cazorla," Wenger told the club's official website. "He's such an important player who turns the game forward, brings movement into the game, makes a game look simple and you feel when he has the ball that something can always happen. Not many players have that in their locker."
Rosicky has sometimes struggled to perform at Arsenal but Wenger credits a tactical change to his strong performance on Saturday.
"Before he played too far away from the box. I always played him wide but I put him in a position where he is more central now. He's more dangerous to score," Rosicky said. "I said to him 'today is a really special game for you because you scored two'. The most important thing is he scores them when you need them and we needed them today."