Paulsen, 21, has been capped by New Zealand at under-17 and under-23 level.
He said: “I’m humbled to be considered worthy of signing with AFC Bournemouth in the Premier League.
“It means the world to me, my family and to my friends to be offered an opportunity to play in the best league on the planet, which most aspiring young footballers only dream of.”
AFC Bournemouth chief executive Neill Blake said: "We're very pleased to secure Alex's services. He's an exciting prospect with so much potential, and his development has been clear for all to see during an excellent season in the A-League.
"We look forward to welcoming Alex to Bournemouth in early July, where we plan to introduce him to Andoni and the rest of the team before he departs for what will be a great opportunity to represent his country at the Olympics."