The footage from the awkward moment, which went down on Tuesday at CinemaCon, has finally surfaced ... and lo and behold, Olivia's handling of it all was as professional as everyone described it.

Check it out ... OW was right in the middle of hyping her new movie, "Don't Worry Darling," and about a minute or so into her speech -- somebody comes up to the stage and drops a manila envelope at her feet. She pauses, asks if that's for her ... and then picks it up.

She openly wondered if it was a script, and playfully joked that she was going to open it right then and there to unpack the mystery -- which she did. As soon as she saw the contents, Olivia simply said, "Oh okay, got it" ... and carried on like nothing had happened.

Olivia held on to the envelope -- marked "personal and confidential" -- for a good while as she finished her monologue, and eventually ... tucked it behind her.

As you know, it was later revealed the papers were custody docs from Sudeikis -- but he denied knowing they'd be served in the manner and time that they were, putting the blame on the process server company. The issue the docs addressed was in relation to jurisdiction.