The Polish champions beat their Scottish counterparts 2-0 at Murrayfield on Wednesday to secure a 6-1 aggregate victory in the third qualifying round for Europe's premier club competition.
But Bartosz Bereszynski's 88th-minute substitution has resulted in Legia's issues.
The defender had a three-game suspension hanging over him heading into the season, with Legia believing two to have been served in the second qualifying round against St Patrick's Athletic.
However, he was not registered for the games, and thus they did not count towards the ban.
As a result, Celtic were handed a 3-0 'walkover' win to take the tie on away goals and are slated to play Maribor of Slovenia in the play-off round.
But in an open letter, signed by chairman Dariusz Mioduski, Legia have asked to meet Celtic in a one-off match in either Warsaw or Glasgow to conclude matters on the field.
After recalling proud moments in Celtic’s history, such as their 1967 European Cup win, the letter read: "I appeal to men, that you have established to the best traditions of honour and integrity… the 126 years were characterised by your great club.
"Do not destroy the beautiful heritage that you have left in the care of previous generations, 'The Bhoys'.
"I challenge you, would you in the spirit of the game and fair play, and on the basis of Article 34 paragraph 5 of the UEFA Disciplinary Regulations… meet in Warsaw or in Glasgow and let's settle this matter honourably."
Speaking before the open letter was published, Mioduski acknowledged a mistake in the director of sport's department at his club but perceived a lowly standing in the eyes of UEFA to be the root of Legia's problem.
"Our Director of Sport is responsible for the department where the error was made. Decisions on their future to be made," he said.
"We still have not received a full written judgement of the UEFA ruling. We are still waiting for it.
"Undoubtedly, our reputation at UEFA is poor. We need to build it in order to be treated fairly. We have no friends there.
"We request our best from our staff but understand a human error. We want Legia to be a place where people want to be.
Legia co-owner and president Boguslaw Lesnodorski was even more damning of UEFA's conduct, alleging the organisation's disciplinary process on the issue went against its own procedures.
"There is an Article 34 Paragraph 5 in the UEFA regulations that allows a fair-play behaviour in such circumstances," Lesnodorski said.
"We were not allowed to present our position to UEFA. In our view, it's a breach of the rules.
"We were not given enough time to prepare our defence line based on the UEFA regulations. We had no opportunity to deal with it.
"The decision was made by one person, not three as required by UEFA regulations. It's a breach of their rules, if we're to be so strict."
