Liel Abada, twice, and substitute Daizen Maeda squandered excellent chances and Callum McGregor struck the post before goals from Vinicius Junior and Luka Modric proved pivotal. Eden Hazard gave the Spaniards a third that was cruel on the hosts.
Ange Postecoglou had vowed his Celtic side would test their intensive attacking style against the European champions and they showed they could pose problems against a starting line-up that featured 10 of the players that began last season's final against Liverpool.
A banner in the Celtic standing area declared they were 'back with a bang' and the hosts had a great opportunity to take the lead inside 30 seconds of their first Champions League group game in five years. The ball fell for Abada inside the box but Thibaut Courtois pushed the Israel international's shot wide at full stretch.
Vinicius was breaking well down the left for Real and Cameron Carter-Vickers blocked Modric's shot but Celtic continued to press and Abada soon spurned an even better opportunity.
Jota's first-time pass sent his fellow winger clear after Celtic took a quick free-kick on the edge of their box, but Abada shot straight at the goalkeeper.
The Scottish champions kept up the pressure. Courtois parried Reo Hatate's powerful volley before McGregor's first-time strike shook the frame of the goal but bounced to safety off the inside of the post.
Federico Valverde fired one close from 20 yards at the other end just before injury forced off Karim Benzema, who was afforded a warm ovation from the home fans. Hazard came on.
Real dominated possession in the latter stages of the half and Hazard stabbed wide from six yards after getting on the end of a Modric cross.
The Belgian soon sent Vinicius clear despite major suspicions of offside but Joe Hart did well to block.
Maeda replaced Abada at the break but missed the best chance yet within two minutes when he scuffed an effort straight at Courtois from six yards following Josip Juranovic's cross.
Celtic paid the price for the misses in the 56th minute. Valverde made space down the right wing and spotted Vinicius cutting in from the left. Juranovic could not make up the ground to cut out the low cross and the 22-year-old slotted into the corner.
The double whammy was complete after Hazard broke clear following Carter-Vickers' challenge to leave Celtic outnumbered. He fed Modric, who cut inside and stroked the ball into the corner of the net with the outside of his boot.
The visitors had taken the sting out of the game and Kyogo Furuhashi's introduction in the 72nd minute lifted the home crowd.
Hazard soon silenced them, briefly at any rate, when he netted from six yards after Dani Carvajal had cut the ball back following a deep cross.
Furuhashi, Jota and substitute Sead Haksabanovic went close to rewarding the Celtic support's incessant backing with a goal but their appreciation of their team at full-time was further evidence that they still believe they can make their mark on the tournament.