After failing to find the net in the opening half-hour, Feyenoord took the lead in the 34th through Brazilian forward Igor Paixao.
Yankuba Minteh and David Hancko swiftly added their name to the scoresheet to give the hosts a commanding half-time advantage.
Not content with their three-goal advantage, Feyenoord continued to press forward, with Minteh grabbing his second before Quinten Timber got in on the act.
There was still plenty of time for Paixao to wrap up a brace of his own and round off a memorable day for Arne Slot's side.
The win moves Feyenoord back to within nine points of leaders PSV Eindhoven, while Ajax are left sitting in sixth position.
Feyenoord threatened to break the deadlock in the opening two minutes when Santiago Gimenez saw his strike kept out by Geronimo Rulli at the near post.
Rulli pulled off another stop in the 32nd minute, rushing out of his goal to thwart Gimenez after gifting possession away with a poor goal-kick.
Just moments later, the Ajax goalkeeper was unable to prevent Paixao from finding the all-important breakthrough.
The 23-year-old collected Minteh's neat through pass before calmly lifting the ball over the onrushing Rulli to spark scenes of jubilation in Rotterdam.
Minteh quickly doubled Feyenoord's advantage, sweeping the ball home from Bart Nieuwkoop's cut-back to make it two goals in as many minutes.
Matters got worse for John van 't Schip's side when Hancko headed the ball in at the near post after latching onto Gimenez's acrobatic attempt.
Van't Schip subsequently made a double change at the break, but any hopes of an unlikely comeback were quashed by Minteh's 56th-minute effort.
The Newcastle United loanee created just enough space to guide a sublime left-foot strike out of the reach of a helpless Rulli.
The raucous home faithful were celebrating for a fifth time when Timber won back possession before drilling a low shot into the bottom corner from the edge of the box.
Paixao then netted his second and Feyenoord's six after Ajax's failure to clear their lines presented him with the chance to thump the ball in off the underside of the crossbar.
Feyenoord were denied a seventh when Alireza Jahanbakhsh saw his effort chalked off for offside, but it did very little to dampen the mood as the hosts celebrated an unforgettable victory.
Slot's side are now unbeaten in 15 Eredivisie matches, while Ajax are left to reflect upon their first-ever loss by a six-game margin in an official game.