The charity, founded by Dr Haruhisa Handa, is committed to developing international sporting, cultural and humanitarian programmes.
Carter, who has begun a two-year contract with Japan's Kobe Steelers, put his left foot to accurate use left, right and in front of the posts.
The balls disappeared towards the northern motorway.
Key responded with a series of elegant lobs from the same spots, proving his handicap has not suffered since resigning as prime minister in December 2016.
He successfully executed one shot while ignoring a heckler bellowing "Jacinda". However, there was disappointment when he "pulled one left" trying to drill a wedge between the uprights from the opposition 22.
Key posed for a three-way handshake afterwards, mimicking the act between himself, then International Rugby Board chairman Bernard Lapasset and All Blacks captain Richie McCaw during the presentation of the World Cup in 2011.
Carter delivered a ceremonial passing of a bag of balls to Westlake students to kick off his role.
The pair conducted their duel before a throng of media - and schoolboys peeking through classroom windows.
"You do get some opportunities to give back to local communities through the All Blacks in South Africa and Europe, and I've got more time at the end of my career," Carter said.
"This gives me a platform, and something as simple as a rugby ball can help inspire and take people's minds off problems in their lives."
Carter said he was impressed how much Key held his nerve in the circumstances.
"I actually thought he might be put off by the cameras, but then I forgot he plays [golf] with [former United States president Barack] Obama with cameras everywhere, so he was pretty relaxed.
"I normally take an hour or so to warm up before kicking. I didn't have that today, but luckily they flew down the middle."
"The aim is to get into communities and give away balls so kids are encouraged to have a play around in the back yard," Key said.
"If you can keep youngsters active and involved in sport, they learn life skills, commitment to a team, and have some fun.
"And let's be honest, obesity is an issue in New Zealand and the Pacific. Part of solving that is education about food and drink - and part of it is exercise."
