The 21-year-old, who joined from Legia Warsaw In 2015, leaves north London without having made a single Premier League appearance for the club, with his only two outings in a Gunners shirt coming in the Carabao Cup.
He enjoyed a successful loan spell at Charlton Athletic last season, helping the club earn promotion to the Championship via the play-offs.
The Gunners were keen to retain the Poland Under-21 international but he favoured a departure from the club if he was not going to be given the first-team opportunities he craved under Unai Emery this season.
He could make his Rams debut in their Championship season opener against Huddersfield Town at the John Smith's Stadium on Monday.
"He is a young player at the age of 21 and it is a permanent deal which I think is a very good investment for the club. I am very happy he has joined the team," Derby manager Phillip Cocu told the club's official website following the signing of Bielik.
"He will get better and develop, but for his age he is a very good player."
The deal gives Arsenal a timely cash boost with under a week to go until the transfer deadline.
The Gunners smashed their transfer record to sign Nicolas Pepe from Lille in a £72m ($87m) deal on Thursday but still hope to add further players before the deadline.
The north London club are still keen on landing Celtic left-back Kieran Tierney, having already seen two bids rejected by the Scottish champions for the 22-year-old.
There are likely to be further departures too with Laurent Koscielny expected to seal a move back to France and the club open to offers for Shkodran Mustafi.