The Gunners beat a number of clubs, including Bundesliga side Hamburg, to the signature of the 17-year-old.
The versatile Poland youth international, who can play in holding midfield and at centre-back, has made just six first-team appearances for Legia in his inaugural season as a professional.
Bielik??is expected to work initially within Arsenal’s Under-21s side, with a view to??a swift progression through to the senior squad.
Wenger said: “He’s a defensive midfielder, he is 6ft 2ins, he has a good energy level and a good technical level. Is ??2m a lot of money? It is less than that but yes, it is a lot of money for a player if he doesn’t succeed. If he does succeed, it’s cheap.
“Of course, it’s a gamble. But on what we saw, it’s a gamble worth taking.”