The 18-year-old flew into Tyneside on Wednesday evening before completing a medical and putting pen to paper on a five-year contract running until 2031. He will wear the number 14 shirt and becomes Newcastle United's third senior signing of the summer, following the arrivals of French goalkeeper Ewen Jaouen and Ivorian winger Bazoumana Toure.
"It's an incredible feeling to be here," Steur said. "It's a giant club in the Premier League and it was always my dream to play in the best league in the world, so this is really exciting. I was at home at Ajax - I joined when I was seven and I leave with only good memories, but when a club like Newcastle comes for you, it's really hard to say no.
"The last few days have gone fast and I'm very happy to join. I'm a player who likes to get on the ball and play forward all the time.
"I'm happy playing between the lines and I feel I have a lot of energy in my game. I just want to win."
Toon manager Eddie Howe welcomed the signing of a player he believes can develop into a mainstay of his midfield.
"We're delighted to welcome Sean to Newcastle United - he's a top young prospect that has already tasted Champions League and top-flight Dutch football," Howe said. "We see real potential in Sean and believe he has the characteristics to become a valuable player for us for years to come. Sean is an exciting addition to our squad. He's had an excellent upbringing at Ajax, who have an outstanding track record of producing young players."
Sporting director Ross Wilson added: "I'm thrilled that we've managed to conclude this deal so swiftly and make Sean a Newcastle United player. We've been aware of his talent for some time but so has the rest of European football, so it was vital that we were able to act diligently and give ourselves the best chance of getting Sean on board."
The move comes after Newcastle sold Anthony Gordon and Sandro Tonali for a combined fee in excess of £160m this summer, and the club remain in advanced talks with Freiburg over a separate deal for midfielder Johan Manzambi.