Leipzig striker Benjamin Sesko scored in the second minute but the linesman raised his flag for offside.
Replays showed Sesko was clearly onside but VAR did not overturn the decision.
Another Leipzig player, Benjamin Henrichs, was offside and despite not touching the ball, was adjudged to have been in the goalkeeper's line of sight.
Madrid's Brahim Diaz scored a stunning solo goal early in the second half to decide the first-leg match.
While praising Diaz's "dream goal", Leipzig coach Marco Rose told reporters the decision was wrong.
"There was a goal which should have counted," Rose said, explaining "if we were playing a goal up, maybe things would have turned out differently".
Rose found some unlikely support from Madrid midfielder Toni Kroos, who agreed the goal should have stood.
"He called it offside because he said he obstructed him (Real's goalie).
"The goalkeeper would not have got to the ball, so it was a goal."
Madrid coach Carlo Ancelotti told reporters "you have to ask the referee" about the controversial call, explaining, "I think he said the goalkeeper was bothered by the (Leipzig) player".
Asked if referee Irfan Pelito, who was taking charge of his eighth match in the Champions League, had shown his inexperience in making the call, Rose shot back "I'm not angry with the referee - I don't want to start a refereeing debate.
"It was a mistake. I think we all saw it was not the right decision.
"I also make mistakes - sometimes two or three per day.
"What's important is that you fix them - and that you try and do it better next time."
Rose said his team still had a good chance of progressing to the quarter-finals, saying "we want to show ourselves in Madrid".
"We want to be the best we can for Madrid, we want to be the most difficult we can be."
