St. Juste scored from a corner in the last minute for Feyenoord, who had been reduced to 10 men in the closing stages.
Napoli, who drop into the Europa League, had made a blistering start in their bid to stay in contention with Piotr Zielinski opening the scoring in the second minute.
The Italian club’s hopes of snatching second place in Group F however were hit as Feyenoord fought back to equalise through Nicolai Jorgensen before claiming the dramatic winner and their first points in this year's group stage.
Napoli finished third in the group with six points, three ahead of Feyenoord and six behind second-placed Shakhtar Donetsk.
Napoli had to win, and hope that second placed Shakhtar lost in their home clash with group winners Manchester City. Yet after a positive start the fight seemed to go out of Napoli when the Ukrainians went 2-0 up against City in the first half of their match, which finished 2-1.
Dries Mertens and Marek Hamsik both had chances to increase Napoli's lead before Feyenoord levelled on 33 minutes as Steven Berghuis's cross was headed into the net by Jorgensen.
Tonny Vilhena was sent off for a second booking in the final five minutes but Feyenoord forced a corner in the closing stages from which St. Juste found the net.
(Reporting by Mark Gleeson; Editing by Toby Davis)
