United, which scored two late own goals in last week's first leg to leave the tie all square at 2-2, produced another desperate defensive display as they went down 5-2 on aggregate at Roman Sanchez Pizjuan.
Once again, Ten Hag's side gifted Sevilla two goals with Youssef En-Nesyri helping himself to a brace after errors from Harry Maguire and David de Gea.
United was without several key players, including the suspended Bruno Fernandes and the injured Lisandro Martinez and Raphael Varane, but Ten Hag insisted that was no excuse for the visitors' display.
"We have to do better, that's the demand. We were not composed, not calm," he said. "We didn't beat the press; when you do, there are so many spaces behind, and it was obvious at the start how to do it.
"We lost the battles, they had more passion, more desire, more willingness. That's difficult to win games.
"It's about the players on the pitch. They have to perform, I believe in them and trust them, but they have to show it, and they were not good enough.
"We have shown on so many occasions good things, but tonight we weren't ready for the game. At this level, playing for Manchester United, you have to be ready for every game.
"This was a great opportunity, a great occasion, to win something, and we gave it away – we have to blame ourselves.
"It's gone, we can't change it. We have to look forward to Sunday [against Brighton and Hove Albion in the FA Cup semi-finals], that's the next opportunity."
Christian Eriksen also accepted United were not at the races as they squandered the opportunity to return to the semi-finals for the third time in four seasons.
"A lot of things happened. We didn't really give them a game from the first minute, and we gave them a lot of opportunities to finish us off," the midfielder said.
"We lost a bit of our coolness, we made more mistakes than we usually do, and the whole team could not catch up with mistakes our team-mates made.
"If you give away goals like we did, you lose, it's nothing to do with the atmosphere. Our own game was not strong enough. It's football, sometimes you have good days, sometimes you have bad days, and it was a bad day.
"Every football player makes mistakes, they punished us straight away. It's how we bounce back, and today we were not strong enough to bounce back from the mistakes."