The defeat leaves second-placed Ajax on 43 points from 21 games, 12 behind leaders PSV Eindhoven and with their championship hopes in tatters.
Former AZ defender Niklas Moisander received a red card just past the hour-mark after colliding with Steven Berghuis and the visitors made their numerical advantage count as American midfielder Johannsson netted having been teed-up by Simon Poulsen.
Ajax have taken a single point from their last three matches, while the victory takes AZ to within one point of PEC Zwolle in third.
Groningen twice came from behind to force a 2-2 draw at relegation-threatened Heracles.
Brahim Darri made it two goals in as many appearances for Heracles since joining from Vitesse last week, but Michael de Leeuw equalised for the visitors midway through the second period.
Wout Weghorst had Heracles in front again and they came within two minutes of three precious points before Tjaronn Chery scored for the fourth game running to rescue a point for Groningen.
(Reporting by Nick Said; editing by Justin Palmer)
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