Making just his seventh appearance since crossing from Montpellier on transfer deadline day, Stambouli fired home from close range after Roberto Soldado's chip had hit the post.
And the French midfielder's 49th-minute strike proved enough for Tottenham to secure their spot in the competition's last 32 on a night where three separate pitch invasions saw the match temporarily suspended in the first half.
Unconfirmed reports suggested that all three invaders were wearing similar shirts promoting the same company.
Play resumed after a short break but Spurs could yet face sanctions from UEFA following the numerous interruptions.
Tottenham's qualification from Group C means top spot will be up for grabs when they visit Besiktas on matchday six. Spurs are two points ahead of the Turkish side.
Partizan, who were already eliminated prior to this game, round off their campaign with a home clash against Asteras Tripolis - who drew 2-2 with Besiktas on Thursday.
Goalkeeper Hugo Lloris returned from a Europa League suspension for the hosts, but Federico Fazio was missing due to a ban of his own.
Aaron Lennon was given a start in his 400th career appearance as Soldado led the line - the Spaniard dragging an effort just wide in the first two minutes.
Tottenham were sluggish in the first half, though, and Partizan - who had not won in their past 16 UEFA club competition group-stage matches - caused them occasional problems on the break.
Petar Grbic stormed down the right and saw his cross clip the shoulder of striker Petar Skuletic, while Danko Lazovic chested Sasa Ilic's hooked ball down before shooting narrowly wide under pressure in the 18th minute.
Soldado just missed Erik Lamela's ball across the face of goal but the most animated the White Hart Lane crowd got during a lacklustre beginning was after two pitch invasions in quick succession.
The situation descended into farce after the third, though, and once play finally resumed, Lamela fired wide in the 11th minute of stoppage time to end an elongated first half.
Spurs did not have to wait long to take the lead after the break as Stambouli - who played Soldado in - pounced after the striker had been denied by the woodwork.
It just was not Soldado's night - he was one-on-one with Partizan goalkeeper Milan Lukac - but could only shoot straight at the onrushing Serbian, while Lamela missed the target on two further occasions.
The competition's joint top-scorer heading into the match - Harry Kane - was introduced as Tottenham head coach Mauricio Pochettino sought to kill the game off with a second goal.
Kane quickly produced a powerful free-kick, only for Lukac to be equal to the task, and the England Under-21 forward also had an effort correctly ruled out for offside with seven minutes left.
Spurs then had Lloris to thank as, with four minutes left, the Frenchman made an outstanding reaction save to keep out a header from the unmarked Vladimir Volkov.
And the stop proved crucial as Stambouli's goal sealed Tottenham's European progression - despite Lamela fluffing his lines and shooting wide after the Frenchman strangely elected to pass rather than shoot in the dying stages.
Lamela also saw a curling effort tipped over by Lukac, but it mattered little.
