Ajax needed a miracle to qualify for the UEFA Champions League knockouts, with as many as eight results elsewhere having to go their way, excluding a win over Olympiacos.
Olympiacos, meanwhile, would only need to beat Ajax to secure a ticket for the knockouts, while a draw would also need results elsewhere to fall in the Greek giants' favour.
Ajax were the first to create a chance, with Davy Klaassen failing to put the ball in the net and instead hitting his own leg, squandering the first big chance for the hopeful home side.
A small handful of chances followed before Oscar Gloukh had the 1-0 on his shoe, but fired the ball over Konstantinos Tzolakis' goal in the 25th minute, keeping the sides balanced despite Ajax controlling possession.
Ajax were left frustrated before half-time after being denied a penalty first, and right-back Anton Gaaei, who formed a sizable threat on Olympiacos' half, lost possession following a promising pass from compatriot Kasper Dolberg, taking two heavy touches to spoil what should've been a big chance for the Amsterdam giants.
Olympiacos upped the pressure following the break and rewarded themselves six minutes into the second half via Gelson Martins, who easily found his way through the Ajax defence and placed the ball in Jaros' far corner for the lead.
The two sides exchanged jabs in a feisty second half, with Gelson Martins and Sean Steur coming closest before the latter struck a Greek hand and appealed for handball. Referee Francois Letexier was called to the screen and awarded Ajax a penalty, which Kasper Dolberg took and fired past Tzolakis, giving his team a late lifeline.
Ajax squandered that lifeline, however, when they left Olympiacos midfielder Santiago Hezze wide open during a corner, allowing the Argentinian to head the ball into the bottom corner to restore the visitors' lead.
With elimination looking very likely already, Wout Weghorst, who replaced Kasper Dolberg, came eerily close to the equaliser but saw his effort get cleared off the line to leave Ajax trailing late in the second half.
The win in Amsterdam completed a magnificent turnaround for the Greeks, who ended on 11 points, mainly thanks to three wins in their final three games.
Ajax, meanwhile, are knocked out of the UEFA Champions League after finishing on just six points and in a disappointing 32nd place.
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