The Germany international has reportedly cost the Premier League club over £40 million, smashing their previous record for a transfer - the £15m signing of Andrey Arshavin in February 2009.
And he has expressed excitement over the prospect of playing in the English top flight.
"I am thrilled to be joining a club of the stature of Arsenal and am looking forward to playing in the Premier League," he said.
"It will be great for my own personal development as a player and I am particularly looking forward to working with Arsene Wenger. From our negotiations it is clear the club has huge ambition and I look forward to being part of an exciting future."
Manager Arsene Wenger also revealed his delight with completing the acquistion of the German playmaker.
"We are extremely pleased to have signed Mesut Ozil. He is a great player, with proven quality at both club and international level. We have watched and admired him for some time as he has all the attributes I look for in an Arsenal player.
"I have said throughout the summer that we have been working hard to bring in top quality players to this football club. This has involved many many people across the club and I have always had the full support of the club in making decisions on the football side.
"Mesut is a top-quality footballer who will be a fantastic addition to our talented squad. We are all looking forward to him joining up with us after his international matches with Germany and working with us as an Arsenal player."
Arsenal have been repeatedly frustrated in the transfer window, seeing moves for Luis Suarez, Gonzalo Higuain and Jesus Navas fail - but Arsene Wenger has finally bolstered his attacking ranks.
Ozil's arrival adds to just two other additions in the window - Yaya Sanogo from Auxerre and Mathieu Flamini returning to the Emirates Stadium from Milan.
He featured in Madrid's first two games of the season, but was visibly unhappy at being substituted in last Monday's 1-0 win over Granada - reportedly going on to spend the rest of the game on the team bus rather than watch his team-mates.
Carlo Ancelotti also revealed Ozil was unhappy at the club after looking on from the bench as they cruised to a 3-1 win over Athletic Bilbao on Sunday.
And Gareth Bale's world-record arrival at the Bernabeu left Ozil's chances of regular football in the balance.
The 24-year-old made 159 appearances for Madrid since joining from Werder Bremen in 2010, scoring 27 goals and racking up an impressive 81 assists.
Ozil rose through the ranks at Schalke, before joining Bremen in 2008 registering a superb 15 goals and 49 assists in 87 games at the Weserstadion.
He went on to play a key role for Germany at the 2010 FIFA World Cup as Joachim low's side reached the semi-finals before defeat to eventual winners Spain.
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