With England boss Roy Hodgson watching from the stands, Welbeck, the £16 million deadline day signing from Manchester United, produced a clinical display to set up a 4-1 win which puts the Gunners back on course for qualification from Group D and marked Wenger's 18th anniversary in style.
Welbeck's cool finishes had put Arsenal 2-0 ahead, before Alexis Sánchez swept in a third. The 23-year-old England man then completed a first career hat-trick on 52 minutes.
Welbeck scored a first career hat-trick against Galatasaray
Wenger has no doubts Welbeck can go on and take his game to another level, with Arsenal set to head to Premier League leaders Chelsea on Sunday.
"It is good that slowly he gains confidence and finishes well. Hopefully the hat-trick will help him gain confidence," said Wenger. "It is not just his finishing because he wants to help the team. His team play is good as well, his teamwork and work-rate.
"I believe that, technically, he is very sound, a good passer of the ball.
"Honestly, I knew he was quick but he can be electric when he starts - he has great pace."