The Norwegian international attacking midfielder has agreed to join on loan from Real Madrid until the end of the season.
He becomes the second signing of the January transfer window following Mat Ryan's arrival from Brighton, also on loan.
Odegaard replaces Mesut Ozil after the German left for Fenerbahce earlier this month and will take the No 11 shirt.
“It’s great that we’ve secured Martin to come to us until the end of the season," Arteta said upon the announcement of Odegaard's arrival.
Martin is of course a player that we all know very well and although still young, he has been playing at the top level for a while.
"Martin will provide us with quality offensive options and we’re all excited to be integrating him into our plans between now and May.”
Technical director Edu is delighted to have finally got his man.
“We are delighted to welcome Martin to the club until the end of the season," he added.
"I would like to thank everyone at Real Madrid and Martin’s representatives for their collaboration in making this loan move happen.
"Martin is an exceptional talent and we are strengthening our squad by introducing an exciting offensive player to the club for the remaining months of this season.”