The 23-year-old fly-half - who is Scottish qualified through his grandmother - will also play for Glasgow Hawks when he's not selected for the Warriors.
Kiwi-born Schulte trialled at the club last season for three weeks, having also featured for Marist Albion Rugby Club and Canterbury age grade sides.
And after playing alongside Warriors fly-half Russell - who went to New Zealand on the Macphail Scholarship in 2013 - during his time in Christchurch, Schulte knows the quality of the outfit he has joined.
"Following my trial last season, all has worked out and I'm delighted to sign for Glasgow Warriors," he told the club's official website.
"I try and pride myself on a running and kicking game. Coming over here and training couldn't be more exciting.
"I'm currently living with Rory Clegg so I've been picking his brains about how the team works and what the plays are.
"Finn Russell played over in Christchurch when I was there, so it was good to see a familiar face when I arrived at Scotstoun.
"He is an outstanding player and someone who I want to learn from."
Schulte becomes Glasgow's eighth arrival of the summer, following hooker Corey Flynn, winger Leonardo Sarto, scrum-half Nemia Kenatale, props Jarrod Firth and Djustice Sears-Duru, fly-half Rory Clegg and second-row Tjiuee Uanivi to Scotstoun.
And head coach Gregor Townsend is confident the No.10 will fit well into the Glasgow culture this term.
He said: "Hagen came over to Glasgow in March and spent a few weeks training with us.
"He fitted in really well with our squad and also managed to play a game for Glasgow Hawks while he was here.
"We're delighted that he's accepted the opportunity to join us on a full-time basis.
"He was top points scorer in the Canterbury club competition last season and he has real variety in his kicking game.
"The step up to a full professional environment gives him a great opportunity to maximise his potential and work on all aspects of his game."
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