Celtic produced a rousing European display away to AC Milan but were beaten 2-0 thanks to a late own goal by Emilio Izaguirre and a further effort from Sulley Muntari.
The Hoops, who beat Barcelona in last season's group stage, were resolute at the San Siro, but eventually came undone as the Rossoneri got a huge slice of luck when defender Cristian Zapata's drive in the 82nd minute appeared to be heading wide before it deflected off Celtic left-back Izaguirre and ended up behind Fraser Forster.
The Celtic goalkeeper was beaten again four minutes later despite making a great save from Mario Balotelli's free-kick, as Muntari knocked in the rebound.
Elsewhere in Group H, Lionel Messi moved closer to Raul's Champions League scoring record with a hat-trick in Barcelona's impressive 4-0 victory over Dutch champions Ajax.
Messi is now just nine goals behind the Real Madrid legend's haul of 71 goals after emulating Cristiano Ronaldo's Tuesday treble with three of his own at Camp Nou.
Gerard Pique was also on the scoresheet as Gerardo Martino's side cruised to an opening win that was damaging to the confidence, and goal difference, of Frank De Boer's Ajax.
In Group E, Schalke dominated Steaua Bucuresti, beating the Romanian champions 3-0 at the Veltins Arena thanks to goals from Atsuto Uchida, Kevin-Prince Boateng and Julian Draxler.
Napoli also claimed a 2-1 win, as they held on to beat ten-man Borussia Dortmund at the Stadio San Paolo.
Gonzalo Higuain and Lorenzo Insigne put Napoli 2-0 up and they got the points despite a Camilo Zuniga own goal, but the red mist which descended on Jurgen Klopp and some of his players were the real story.
Klopp is known for his cool demeanour, but he was positively raging as he towered over the fourth official, barking at him relentlessly, in the wake of Higuain's goal.
He was unhappy that Neven Subotic had been allowed back on after treatment but not in time to prevent the goal. Klopp was dismissed to the stands and on the stroke of half-time was joined by goalkeeper Roman Weidenfeller who raced out of his box to try and close down Higuain but handled in front of the referee.
Lucho Gonzalez scored the only goal of the night as Porto ensured Austria Vienna's debut in the Champions League group stage ended in a narrow 1-0 defeat.
Atletico Madrid got their European season off to a strong start by producing a thorough performance to beat Zenit St Petersburg 3-1 at the Vicente Calderon.
Joao Miranda's headed goal had given the hosts the lead towards the end of the first half, but a stunning effort from Hulk drew the Russian side level. Arda Turan restored Atletico's lead not long after the equaliser, and then substitute Leo Baptistao scored his first competitive goal for the Rojiblancos to seal the win.
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