The Magpies have confirmed 24-year-old Holland international Janmaat's arrival and the France international's departure for the Emirates Stadium.
Press Association Sport understands Newcastle will receive around £12million for Debuchy, around half of which has been invested in his replacement.
The Dutchman, who underwent a medical on Tyneside earlier this week, has signed a six-year contract and will replace Debuchy as the club's first-choice right-back.
Janmaat, who has just returned from the World Cup finals in Brazil, said: "Newcastle is a fantastic club, with a big history and a great stadium. I have also been told all about how great the fans are.
"This is the perfect move for me, and I am really looking forward to being with the team and getting started here at Newcastle.
"I think we are going to have a great season."
Manager Alan Pardew was delighted to get his hands on a man the club had targeted since it became clear that Debuchy would be leaving.
He said: "We are delighted to bring Daryl to the club.
"He is the perfect example of a modern full-back: someone who is good defensively, but offers a real threat going forward as well.
"Daryl had an excellent World Cup and we are looking forward to him continuing that form for Newcastle."
Janmaat becomes the club's sixth summer signing with owner Mike Ashley having handed Pardew the financial backing to carry out a major overhaul of his squad.
He is the third new arrival this week following in the footsteps of Montpellier midfielder Remy Cabella and Monaco striker Emmanuel Riviere, and he is unlikely to be the last with the capture of another frontman a priority.
Debuchy's departure just 18 months after his arrival from Lille comes as something of a disappointment with Pardew thinking he would be a long-term signing after reuniting him with close friend Yohan Cabaye.
However, Cabaye's January exit for Paris St Germain and a dreadful second half of the season seemed to unsettle the defender, and it became inevitable that he would leave this summer.
The compensation for Ashley is that he more than doubled his money on the Frenchman, and the proceeds of his sale have effectively covered the cost of recruiting Riviere and Janmaat.
Tweet:Daryl Janmaat won more tackles in the opposition’s half (7) than any other player at the 2014 World Cup. #NUFC
A club statement said: "The club wishes to place on record its thanks to Mathieu for his efforts during his time at Newcastle. He leaves with all our best wishes for his future at Arsenal."
Janmaat, like Cabella, will not travel with the squad on a week-long trip to New Zealand as he enjoys an extended break after his exertions in Brazil.
Nor will midfielder Hatem Ben Arfa, who appears to be facing a bleak future on Tyneside.
The 27-year-old France international has been training away from the main group since his return for pre-season and, according to reports from France, has been fined for arriving back on Tyneside overweight - a charge he has denied - although Newcastle were keeping their own counsel on the matter on Thursday.
- Sports & Recreation