NICOSIA - Inter held Anorthosis to a 3-3 draw despite twice taking the lead in Tuesday's Champions League Matchday 4 clash at the GSP Stadium.
Mario Balotelli and Marco Materazzi had goals cancelled out by Cédric Bardon and Giorgos Panagi in the first half.
Second-half substitute Julio Cruz equalised late on after Nikos Frousos stunned the Nerazzurri with Anorthosis' third goal just after the break.
Inter remain three points clear of the Cypriots at the top of Group B, with Panathinaikos moving up to third spot after their 3-0 victory over Werder Bremen.
Mourinho's men were straight into attack in Nicosia and Albanian goalkeeper Beqaj saved Ibrahimovic's low effort with his feet after palming clear Maicon's cross.
The home side hit back on the counter-attack but Julio Cesar was quick off his line to thwart the onrushing Taher.
Inter were in front inside 13 minutes when Balotelli slotted home his first Champions League goal from near range after Beqaj failed to hold Ibrahimovic's blistering long-range free kick.
Determined to keep intact their unbeaten run at home this season, Anorthosis soaked up a period of pressure and equalised just past the half hour when Bardon pounced on Burdisso's miscued headed clearance before slotting into an open net.
Ibrahimovic and Quaresma had shots saved, with Balotelli hooking just wide as Inter battled to retake the lead, and they did on 44 minutes when Materazzi headed low through Beqaj's legs from Balotelli's corner.
The lead was short-lived as Anorthosis equalised again in added time at the end of the first half. Cambiasso's attempt at a headed clearance from Dobrasinovic went in off substitute Panagi, who came on for the injured Paulo Costa in the 43rd minute.
Anorthosis forward Frousos sent the home fans wild five minutes after the break, firing an angled shot past Julio Cesar after a second mistake by Burdisso, and Inter found themselves with a task on their hands.
Mourinho played all his cards in one go, hauling off Stankovic, Burdisso and Mancini and replacing them with Vieira, Cruz and Maxwell.
Cruz sparked a reaction when his flicked header from Quaresma's cross was kept out by Beqaj. But Ketsbaia's side came looking for a fourth and almost got it in the 69th when Constantinou forced Julio Cesar into a tipped save.
Beqaj made a brilliant two-handed block from Zlatan Ibrahimovic after a mazy run by Maxwell as Inter threw everything into attack. The reward came with nine minutes left when Cruz headed in Balotelli's 30-yard free kick for his thirteenth Champions League goal in black and blue.
Inter created chances for a late winner and Anorthosis had Beqaj to thank for a goalline clearance from Cambiasso's header, before Ibrahimovic miscued in front of an open goal.
Anorthosis 3-3 Inter (HT: 2-2)
Scorers: Balotelli 13, Bardon 31, Materazzi 44, Panagi 45+1, Frousos 50, Cruz 81.
Anorthosis Famagusta: 1 Beqaj; 28 Georgiou, 4 Katsavakis, 24 Constantinou, 12 Leiwakabessy; 27 Bardon (22 Ocokoljic 78), 6 Dobrasinovic, 20 Laban; 19 Taher, 11 Frousos (8 Skopelitis 63), 10 Paulo Costa (21 Panagi 43).
Subs not used: 30 Nagy, 9 Sosin, 16 Tsitaishvili, 23 Samaras.
Coach: Temuri Ketsbaia.
Inter: 12 Julio Cesar; 13 Maicon, 16 Burdisso (14 Vieira 58), 23 Materazzi, 4 Zanetti; 77 Quaresma, 19 Cambiasso, 5 Stankovic (9 Cruz 58), 33 Mancini (6 Maxwell 58); 8 Ibrahimovic, 45 Balotelli.
Subs not used: 1 Toldo, 2 Cordoba, 39 Santon, 36 Bolzoni.
Coach: José Mourinho.
Referee: Laurent Duhamel (France).