PUBLISHED: 11:05 GMT, 27 July 2012 | UPDATED: 12:15 GMT, 27 July 2012
The 23-year-old had been linked with a move to the Gunners’ fierce rivals Tottenham as part of Luka Modric’s proposed move to Spain.
The German-born Turk only joined Madrid from Borussia Dortmund last summer.
Sahin’s capture comes just days after Arsenal boss Arsene Wenger appeared to rule out signing a midfielder this summer.
Responding to speculation that he is keen on Rennes player Yann M’Vila, he said: ‘No [I am not interested in M’Vila]. We wait on [Abou] Diaby and [Jack] Wilshere.
‘When you look we have Alex Song, Aaron Ramsey, Francis Coquelin, Wilshere, Mikel Arteta and Diaby.
‘In defensive midfielders and box-to-box players we are not short if they are all coming back.
‘If we have setbacks in pre-season it could be different.’
Sahin became the youngest player to appear in the Bundesliga when he made his debut for Dortmund in 2005, aged 16. He later became the youngest player to score in the competition.
He signed a six-year deal with Jose Mourinho’s side when he was snapped up for £8million last May but made just 10 appearances in his debut season.
Despite obtaining a German passport last year, Sahin has been capped 31 times by Turkey, and completed a hat-trick by becoming the youngest player to be capped by his country.
Sahin does have history with new Arsenal team-mate Lukas Podolski however.
The pair clashed in 2010 while playing for Dortmund and Cologne respectively but will have to put their differences aside if they are to end Arsenal’s seven-year wait for a trophy.