The Red Devils looked to be on their way to securing all three points when a Rasmus Hojlund brace sent them two goals ahead in the opening 28 minutes of the contest in Denmark.
However, Marcus Rashford was sent off before Copenhagen netted twice in the closing stages of the first period through Mohamed Elyounoussi and Diogo Goncalves.
Bruno Fernandes sent Man United back ahead 20 minutes from time, but Copenhagen netted twice in the final exchanges through Lukas Lerager and Roony Bardghji to secure all three points.
Ten Hag has said that Rashford was unfortunate to be sent off, with the England international penalised for a foul on Elias Jelert following a VAR check when he appeared to have stepped across the defender in a bid to protect the ball.
The Man United head coach had no complaints about Copenhagen's third or fourth goal, but he has said that the first two should not have counted, as the home side's opener should have been ruled offside, while the second came from a debatable handball from Harry Maguire.
"We are very disappointed, because you play very good. I think we started the game so well, the best minutes of this season," Ten Hag told reporters during his post-match press conference.
"We are winning the game and I think the red card changes everything. And then we concede two goals before half-time which should never have counted. The first is offside, the second is a penalty. Four penalties against [in the Champions League] and I'd say three are very debatable.
"It's also harsh (the red card). I think he went for the ball and the referee has such a long time. They need such a long time to make it a red card.
"I think when you freeze it, it looks always so [much] worse. And as I say, it takes them so long and they make a red card of it. I'm very disappointed about such decisions.
"I think the game is never meant to be like this. This has nothing to do with football. Decisions have to be made and I accept that there are also wrong decisions made. If I sum it up, then you make on this level three such tough decisions... you control the game, and I think the game is never meant to be for that.
"Football is a game of mistakes. I will not say that we did everything right, but I take a lot of positives. Even with 10 men we were dictating the game and we played very good. And so yeah, we are very disappointed that in the end that you take nothing from this, because it could have been different."
Man United, who have now lost nine times in all competitions this season, are bottom of Group A on three points, one point behind second-placed Copenhagen.
Ten Hag's side face a must-win game against Galatasaray in Turkey before the end of the month, but they first have Premier League matches against Luton Town and Everton either side of the international break.