Needing just a point against RB Leipzig to seal a place in the last 16 of the competition, United found themselves 3-0 down to the German side after 69 minutes.
A penalty from Bruno Fernandes and an own goal by Ibrahima Konate with eight minutes to go gave United a chance to snatch an unlikely draw and the crucial point they required.
In the final minute, Paul Pogba whipped in a dangerous cross which defender Nordi Mukiele attempted to block but the ball hit goalkeeper Peter Gulacsi before it could go into the back of his own net.
The final whistle sparked jubilant scenes among the Leipzig players and backroom staff while United were disconsolate.
Carragher was on punditry duty for CBS Sports and his reaction to United’s late chance was caught on camera.
Carragher jumped out of his seat and celebrated loudly as United were denied an equaliser.
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After the match, United captain Harry Maguire said the team only had themselves to blame.
“We had belief and confidence going into this game that we were going to go here and get a result,” he said.
“We started the game really poor, went two goals down. And then… for 60 minutes of the game we’re the strong force and much better team. We need a third goal but the third goal didn’t come.
“We knew going into half-time that the third goal was going to be crucial, we said the same on Saturday but today we didn’t manage to get it.
“We had a lot of shots, Bruno’s hit the bar with a great free-kick and on another day that goes in. We’ve got to look at ourselves.
“It’s not good enough for this club to go out in the group in the Champions League. We’re very disappointed.”