The Gunners boss has heaped praise on the 20-year-old midfielder, who has impressed this season since returning from an accumulation of injuries which ruled him out for 14 months
Two penalties from Mikel Arteta sealed a first victory in four league games for the Gunners, with the England midfielder standing out as one of the star performers.
The youngster has impressed since returning from a lengthy spell on the sidelines, and the French manager think Wilshere is close to returning to the all-action performances he displayed before the injury.
"[Wilshere] is getting closer to the player we know he is and today I think he made a big step," he told reporters.
"You could see it already this week in training, that he has gained a little burst again, a little zip in the first three or four yards.
"That's what you have to accept in football, you have sometimes to be patient. He can only gain it by playing but of course in the first games you cannot be at that level."
The Arsenal boss also praised Alex Oxlade-Chamberlain, who won his side's second penalty after being fouled by West Brom's Chris Brunt in the second half: "I believe he is on the same trend as Jack. He is getting better, stronger, more mature, especially more determined today.
"There is much more man in his game than there was before. I am pleased with what I have seen from him today," he added.
Arsenal now sit in seventh position in the table after a poor start to the season, but the result will be a relief for Wenger, who has faced increasing criticism as his side have fallen 15 points behind league leaders Manchester United.