Arsenal failed to qualify for the Champions League for the first time in 20 years despite a 3-1 win over Everton on the final day of the Premier League.
The Gunners began the day a point behind fourth-place Liverpool and were on course to displace the Reds when they took an eighth-minute lead.
But Liverpool, needing a win to ensure fourth, broke Middlesbrough resistance just before the break, and won 3-0.
Manchester City claimed third with a very easy 5-0 win at Watford.
Arsenal’s remarkable run of qualifying for the Champions League in 19 consecutive seasons came to its widely anticipated end on Sunday.
Hope grew among the home support when Hector Bellerin gave the Gunners an early lead and news filtered through that Liverpool were struggling against the relegated Teessiders.
But in first-half stoppage time Georginio Wijnaldum put the Reds ahead with a stunning strike, and Jurgen Klopp’s men eventually ran out comfortable winners.
Arsenal manager Arsene Wenger: “It is annoying but we had a spell during the season that was difficult and it was difficult for me in my personal situation.
“We were playing in a hostile environment. The players came back stronger in the last two months and I’m very proud of them for doing that.”
The Reds will play in the Champions League for the first time since the 2014-15 season.
Liverpool suffered from early nerves at Anfield before Wijnaldum’s opener on the stroke of half-time eased the tension.
Philippe Coutinho added a second after the break with a free-kick, and Adam Lallana slotted in to make it 3-0.
Liverpool manager Jurgen Klopp: “We’ve had bigger games – this was the game of the season though.
“We had this pressure but we are Liverpool and have to deliver.
“Everyone knows where we missed out on points, but we will improve. A club like Liverpool needs to be in the Champions League.
“I’m proud of the boys. We work a lot. We are closer together. We have to build together.”
Manchester City sealed third spot and automatic qualification for the Champions League group stage with an emphatic 5-0 win at Watford.
Pep Guardiola’s side required only a point to secure a top-four place, but instead of sitting back went about dismantling their opponents.
Sergio Aguero scored twice, after Vincent Kompany’s opener, with Fernandinho and Gabriel Jesus also getting on the scoresheet.
City boss Pep Guardiola: “We were under a lot of pressure. Congratulations to Chelsea and to Tottenham, we are so glad to be third.
“It is not a club with history of playing in Europe like Manchester United or Arsenal. But now we are there five or six years and now we can try to close the gap on the elite.
“The best teams in Europe will be at the Etihad Stadium next season. I don’t know what we need to add. This is one of the best groups I have ever trained – they never gave up. It was a pleasure to be with them.”