The Denmark international set up one goal and scored another as the reigning Dutch champion triumphed in Alkmaar, while Groningen also has a 100 percent record after two rounds.
The reigning Dutch champion produced the perfect response to PSV's 6-1 thrashing of NAC Breda on Saturday with victory in Alkmaar 24 hours later.
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Denmark international Schone played a key role in the victory and he set up Davy Klaassen for the game's opening goal in the 15th minute. Schone's ball played in Klaassen and his angled finish from 15 yards separated the two sides at the break.
AZ, which finished eighth last season, drew level five minutes into the second half when Steven Berghuis coolly curled past Ajax goalkeeper Jasper Cillessen with his left foot.
But Schone turned the game on its head with a remarkable goal 21 minutes from time that paved the way for Ajax's triumph.
Schone initially did well to even keep the ball in near the corner flag, and while AZ defender Mattias Johansson was appealing for a throw-in, his opponent skipped past him and began to run across the face of goal.
Schone's run kept going as he arrowed across the pitch before, 20 yards from goal, he unleashed a powerful strike into the bottom right corner.
Ajax, which hosts joint league leader PSV next week, wrapped up the win in injury time when halftime substitute Anwar El Ghazi netted from 20 yards.
Groningen made sure four teams have 100 percent records after two matches with a 3-1 success against Heracles.
Tjaronn Chery and Michael de Leeuw scored in the first half for Groningen, and although Iliass Bel Hassani hit back for the visitors, Yoell van Nieff's 61st-minute effort settled the contest.
Meanwhile, Utrecht left it late in its 2-1 home win against Willem II that saw substitute Leon de Kogel bag an 88th-minute winner.
Robbie Haemhouts canceled out Jens Toornstra's goal with just 14 minutes to play but Utrecht capitalized on a defensive mix-up to claim all three points.
Toornstra's simple ball across the face of goal was poorly allowed to run all the way to the far post, where De Kogel stole in to net from close range.
Sunday's other match saw Sander Fischer's 94th-minute penalty give Excelsior a thrilling 3-2 win against Go Ahead Eagles.