Vandoorne, one of the most highly regarded prospects outside F1, has been with the team since 2013. The Belgian took a record seven victories en route to securing a dominant GP2 crown in 2015, and this year made his Grand Prix debut as a stand-in for the injured Fernando Alonso at Bahrain, where he finished 10th and scored the team's first point of the season.
Button, who says he has no intention to retire, will stay on at McLaren in a new ambassadorial role for 2017 and 2018.
The 2009 world champion is already the third most experienced driver in F1 history, and this year's Abu Dhabi finale is set to be his 305th F1 Grand Prix start - which would move him one short of Michael Schumacher for second on the all-time list.
Executive chairman and CEO Ron Dennis paid tribute to Button, and insisted Vandoorne is 'clearly ready' for the opportunity to race next year.
"McLaren-Honda’s race driver line-up next season will be Fernando and Stoffel – a perfectly balanced mix of proven brilliance and immense potential," Dennis said.
"But, before I speak on the subject of either of them, I want to talk about Jenson. As a race driver for our team these past seven seasons, he’s been superb, both on and off the track. And, as we’re seeing this season, he remains superb – not only fast and fit but also experienced and expert. He’ll start his 298th Grand Prix tomorrow; as such, he’s the most experienced driver on the grid.
"Having extended his contract to include 2017 and 2018, he’ll continue to be a senior, influential and committed member of the team, and will remain centrally involved in the development of our cars. He’ll also be available to race for us if circumstances require it.
"Stoffel has been a member of the McLaren Young Driver Programme since February 2013, and was appointed our Test and Development Driver 11 months later.
"Over the past two-and-a-half years he’s made a very valuable contribution in that role, learning from his two World Champion team-mates as well as from our many world-class engineers, and, having been runner-up in World Series by Renault in 2013 and the run-away winner of the GP2 Championship last year, he’s now completed with distinction a meticulously designed apprenticeship.
"He’s consequently now ready to race in Formula 1, as his fine Grand Prix debut in Bahrain this year proved."
"It’s an enormous honour to become a McLaren-Honda race driver, and I promise to work as hard as any Formula 1 driver ever has before," Vandoorne said.
"I already know the team extremely well – they’re a fantastic group of extremely clever people – and I’ve already raced the car once and tested it often. I have total confidence in the potential of the McLaren-Honda project, and I firmly believe that, racing alongside a brilliant and experienced driver, I’ll be able to make a strong contribution to the team’s future World Championship successes.
"I’m not saying when that’ll happen, but I’m confident that that’s what the McLaren-Honda team will achieve, and I’m determined to do my bit.
"There are a lot of other people whom I’d like to take this opportunity to thank – too many to list in fact. So all I can say is: you know who you are, and thank you."
Alonso meanwhile said he is confident he and Vandoorne can help propel McLaren to the front of the grid next year, adding: "I’ve come across a lot of young drivers over the years, and I know a good one when I see one.
"I enjoyed working with him when he stood in for me in Bahrain earlier this year, and I’m looking forward to continuing to work with him, and to helping him learn and develop. Together with Jenson, I’m sure he and I can take McLaren-Honda back to its rightful place: at the front of the grid."