As revealed by Goal, the French fullback agreed his move to the Emirates Stadium earlier this month but completion of the deal was held up while Newcastle sought a replacement.
However, United has now finalized formalities in the signing of Netherlands international Daryl Janmaat on a six-year deal, allowing Debuchy to sign on the dotted line with the Gunners.
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"We are delighted to welcome Mathieu Debuchy to Arsenal," Arsene Wenger told the club's official website.
"He has shown he can perform at the highest level with his club sides and also for France. He is a quality defender who has good Premier League experience and I'm confident he will fit in very well with us."
Debuchy added: "I'm very proud to be joining a great club like Arsenal and to wear its colors, it's one of the biggest clubs in the whole world. I'm looking forward to working with Arsene Wenger and to helping the team build on last season's FA Cup success.
"Playing again in the Champions League is a big excitement for me and I will do my best to help Arsenal compete for trophies."
Arsene Wenger was search for a replacement for right back Bacary Sagna, who joined Manchester City on a free transfer earlier in July after telling the Gunners who would not extend his contract.
The France international becomes Arsenal second major signing of the summer following the arrival of forward Alexis Sanchez from Barcelona.