Anwar El Ghazi gave the hosts the lead in the first half before a bilstering turnaround from the visitors gave them their first win at the Amsterdam ArenA in six years.
Both teams kicked off with identical records - two wins from as many matches - and in the knowledge that whoever won this encounter would replace PEC Zwolle at the head of the table on goal difference.
PSV enjoyed most of the early possession without endangering the Ajax goal, so it was something of a surprise when the the host took a surprise lead after 15 minutes.
Davy Klaassen on the left picked out the run of Anwar El Ghazi through the middle, and he marked his first ever Eredivisie start by glancing a header over the advancing PSV keeper Jeroen Zoet to give Ajax the opening goal and El Ghazi his second in as many games.
A goal to the good, Ajax took control of the game and pressed to double its advantage. Approaching the half-hour, Boilesen broke forward, his low, left-footed shot palmed out by Zoet as far as Klaassen, who managed to turn and shoot only for Rekik to make a crucial block.
PSV did manage to create a few chances but found Cillessen equal to the near-post attempts from first Georginio Wijnaldum, then Adam Maher.
After the interval, Ajax - chasing a fifth successive domestic title - looked to press home their advantage but were left stunned after being struck not once, but twice, by some deadly counterattacking from PSV.
First, Luciano Narsingh used his blistering pace to outstrip the Ajax cover, sliding the ball square across goal for Memphis Depay to convert with a simple tap in.
Then, after 63 minutes, Lasse Schone's wayward pass in midfield was collected by Luuk de Jong, who drove forward from halfway before playing in former Ajax academy player Narsingh to this time become the goalscorer.
Ajax pressed for an equalizer, and it took a truly brilliant save from Zoet to deny Henrik Moisander who connected with a great pass from Daley Blind, in what many expect to be his final game in an Ajax shirt.
If that proves to be the case, then it will have been a disappointing end to Blind's time at the club - five minutes from time, PSV took the points when Florian Jozefzoon capitalized on some chaotic Ajax defending to fire home a match-clinching third.