Odegaard has agreed to join the Gunners until the end of the season, with the loan deal containing no option for Arsenal to make the move permanent in the summer.
The 22-year-old has made just seven La Liga appearances for Real Madrid this season and he is eager to gain more action in the second half of the season.
Odegaard had also attracted interest from Real Sociedad, where he spent last year on loan, but Arsenal have won the race for the attacking midfielder.
The Gunners have been in the market for a playmaker throughout the current transfer window and have looked at a number of targets.
Odegaard’s availability became clear last week, when Real Madrid decided he would go on loan, and Arsenal’s technical director Edu has acted swiftly to tie up a deal.
Arsenal wanted to sign a creative midfielder this month to provide competition for Emile Smith Rowe, who has burst onto the scene and into the first team.