Defensive blunders by Neil Lennon's men repeatedly let in the Italian legends and they finished clinically to claim an invaluable away margin.
Celtic, watched by former idol Henrik Larsson and Rod Stewart in the 60,000 Parkhead crowd, will now have to travel to Turin on March 6 and score at least three times to reach the quarter-finals.
The Hoops fell behind as early as the third minute when Nigerian Efe Ambrose, who arrived back in Glasgow from the African Cup of Nations final in South Africa only hours before the game, mistimed a through ball and Alessandro Matri scored.
In the second half, as Celtic chased the game, Juventus - marshalled as ever by Andrea Pirlo - soaked up the pressure and scored twice on the break through Claudio Marchisio and Mirko Vucinic, with Ambrose again at fault for the third.
Lennon later admitted Celtic "need a miracle" in the second leg. The 3-0 scoreline effectively ends the tie as a meaningful contest, as Lennon acknowledged.
"We need a miracle," he said. "But it's the harsh reality of Champions League football and some of my young players will learn a huge lesson tonight.
"I thought for 70-odd minutes, until the second goal, we were by far the better side - but you can't give away goals like we gave away, sloppy.
"Efe was poor for the first goal but we had good chances to equalise," said Lennon.
"Then we switched off defensively for the second goal and the third goal was poor."
One legitimate objection Lennon could have was over the way Juventus - and Stephane Lichtsteiner in particular - defended corners. The Swiss player was booked after tangling with Gary Hooper but continued grappling with the striker and also Ambrose and Scott Brown.
"I'd like to ask the referee, is the game different in Spain or Italy?" Lennon said. "Every time one of my players tried to move he was held. He should have given a penalty on at least two occasions."