Atletico Madrid advanced to the quarterfinal stage of the competition at Man United's expense following a 1-0 win at Old Trafford.
Renan Lodi scored the only goal of the game with a header in the 41st minute, which saw Atletico qualify 2-1 on aggregate.
Rangnick took issue with Atetico's goal for what he believed was a foul on Anthony Elanga in the build up, and the German was also unimpressed with constant breaks in play in the second period as United chased an equaliser.
"Against a team like Atletico, it is highly important you score the first goal yourself," said Rangnick.
"If you don't do that, it is even more important not to concede a goal.
"One minute before half-time, we conceded that goal in a transitional moment.
"I believe there was a foul on Anthony Elanga before that but the linesman and referee did not see it that way - I think we shouldn't have been 1-0 down at half time, I think we were the better team in the first half."
Rangnick added: "In the second half, I don't know if the game was played for more than two minutes without somebody lying on the floor.
"I think there were some curious refereeing decisions. I wouldn't necessarily say they were decisive, maybe the one before they scored, but I think the referee made it pretty easy for Atletico to do their time-wasting antics and that didn't make it any easier for us.
"It was frustrating and difficult for us in the second half. From normal play, we didn't produce that many chances."
