The player had a medical at Arsenal last week and moves to the Emirates Stadium in a deal that British media reports have put at around two million pounds ($3.02 million).
Bielik, who has been capped for Poland's under-17s, signed for Legia from their arch rivals Lech Poznan in July.
He has made six appearances in all competitions for the capital club this season.
Arsenal manager Arsene Wenger said last week that the teenager, who can also play in defence, will go straight into the first team squad on his arrival.
"It's a gamble but, on what we saw, it's worth to take the gamble," Wenger said.
"It's an awful lot of money for a player if he doesn't succeed. If he does succeed, then it's cheap." ($1 = 0.6617 British Pounds)
(Reporting By Sam Holden, editing by Alan Baldwin)
