Real head coach Carlo Ancelotti admitted his side struggled in the first half at Celtic Park but the Scottish champions missed a number of chances and ended up losing 3-0.
Liel Abada and Daizen Maeda both had glorious opportunities to open the scoring and Hoops captain Callum McGregor hit the inside of a post before second-half goals from Vinicius Junior, Luka Modric and Eden Hazard saw the Spaniards prevail.
Celtic now face Shakhtar Donetsk in Warsaw next Wednesday. The Ukrainian side are fresh from a 4-1 away win over Leipzig - with the help of a double from former Hoops winger Marian Shved - which swiftly prompted the German club to sack head coach Domenico Tedesco.
Postecoglou came away from the opening defeat with a resolve to do better.
"I feel determined, not optimistic," the manager said. "I feel that responsibility, I want us to be a Champions League football club. For that to happen we have to come out next week and be more clinical.
"It's not about being optimistic, it's about being determined to bridge that gap.
"Every game is going to be hard, it's the Champions League. I don't think the next game is going to be any different. There's going to be fine margins and we just have to make sure we end up on the positive side of the ledge."