The French defender moved to the Emirates Stadium last month following a brief spell with the Magpies.
"It was the right time for me to make this big step forward and this big move," the 29-year-old told the club's official website.
"When my agents told me Arsenal were interested, I said yes straight away! There was no question, no debate, no doubt, nothing! You don't turn down a club like Arsenal."
Debuchy won his first piece of silverware with the Gunners last Sunday when they defeated Manchester City 3-0 in the Community Shield.