They had won it three times in a row, four times in the last five years, while the last time Madrid failed to reach the quarter-finals was 2010, in Cristiano Ronaldo’s first year at the club.
Ronaldo rescued them last season in the quarter-finals and then Gareth Bale won them the trophy, but the former is gone and the latter started on the bench here, dropped again by Solari.
Solari oversaw an improvement after Julen Lopetegui was sacked in November but his team, 12 points behind Barcelona in La Liga, have watched their season unravel in a week.
Madrid president Florentino Perez is now faced with deciding whether it is worth keeping his coach until the end of the season, let alone beyond the summer.
‘I did not come to the club in such a difficult moment to give up,’ Solari said afterwards.
He could have few complaints. Ajax were hard-done-by in the 2-1 first-leg defeat and dominant at the Santiago Bernabeu, their scintillating victory confirming a 5-3 win on aggregate and a place in the quarter-finals. On this evidence, they can go further.
Hakim Ziyech and David Neres put them two up at half-time and the excellent Dusan Tadic made it three before Marco Asensio gave Madrid hope. It lasted two minutes, as Lasse Schone sent a bending free-kick into the top corner before Nacho was sent off in injury-time.
‘The week has been hard for us,’ said Nacho. ‘We have practically said goodbye to the league, the cup and today was like a final in the Champions League.’
‘I’ve never felt as sick as this, I don’t know how to explain it,’ said Dani Carvajal.
Bale was on after 29 minutes for the injured Lucas Vazquez but hit the post and then missed two good chances. Sergio Ramos was in the stands, having earned a deliberate booking to clear a ban, believing this tie was already won.
Raphael Varane might have finished it within five minutes but headed onto the crossbar from six yards and instead Ajax took control.
Toni Kroos, poor all night, was caught in possession from a throw-in and Tadic skipped free down the right. He pulled back for Ziyech, who guided it with his left foot into the far corner.
Ajax were bouncing as Neres baffled Carvajal with a pirouette in the corner and then Tadic did the same, spinning around Casemiro in midfield before teeing up the second. Neres rounded Thibaut Courtois and chipped the ball home.
Vazquez and Vinicius Junior both left the field injured and in tears, the latter after an electric run forward. Bale came on, with Asensio, and raced down the left but his finish from the angle clipped the post.
The fans whistled at half-time and Madrid were no better after, creating chances but lacking composure and control. Karim Benzema shimmied past Ziyech but curled wide, only for Ajax to add a sumptuous, albeit controversial, third.