Renan Lodi scored the only goal just before half-time as Diego Simeone's side won 2-1 on aggregate after a late draw in the first leg last month.
Tuesday's result means United's trophy drought will now almost certainly stretch to five years, with Ralf Rangnick's side already eliminated from the FA Cup and 20 points behind leaders Manchester City in the Premier League.
"I think we played a very good first half -- exactly the way we wanted to play with a high energy level but we couldn't convert that into one or two goals," Rangnick said after the match. "Conceding that counter-attacking goal just before halftime didn't make life any easier. It was hard in the second half and always interrupted. There was always somebody lying on the floor.
Teenage United forward Anthony Elanga, who turned hero to score a last-minute equaliser in the first-leg, went close to opening the scoring on Tuesday when his run toward the near post on 12 minutes was rewarded with a low cross from Bruno Fernandes, but he saw his shot stopped by goalkeeper Jan Oblak.
- Ogden: Manchester United, the club where big stars go to fail
- Stream ESPN FC Daily on ESPN+ (U.S. only)
- Don't have ESPN? Get instant access
It was a typical display of the kind of gritty defending and swift counter-attacking for which Simeone's side have been known while United, for all their pressure, were unable to create enough quality opportunities to get themselves the goal they needed.
The visitors did eventually take the lead on 41 minutes through Lodi as he scored a jumping header at the back post following a floated cross from Antoine Griezmann. The move came after Atletico won back possession from Elanga, who looked to have been fouled but saw protests waved away by referee Slavko Vincic.
On Vincic's calls, Rangnick added: "I would also say some curious refereeing decisions. I wouldn't say they were decisive but at least he fell too often for those time-wasting antics."
"I'm really happy for the opportunity that the coach gave me to play out of my position," said Lodi, a defender who played as a winger for the game. "I'm a left back, that's what I do best, but if I'm needed in attack like the last two matches, I'll do my best to contribute. I'm delighted that I finally had the chance to show what I'm worth."
United pushed for an equaliser in the second half but were unable to find a way back into the match, resulting in their sixth elimination in their past eight Champions League knockout ties. After the final whistle blew, Simeone sprinted towards the tunnel and had plastic bottles, drinks and other objects thrown at him.
"In the second half it was a brilliant collective effort from the team, one of the best this year," Simeone said in a postmatch news conference. "I'm happy, that's why I celebrated it in the dressing room."
Information from Reuters contributed to this report.