Despite a comfortable victory over Everton on the final day, the club missed out on a place in the European Championship for the sixth year in a row.
With Spurs winning at Carrow Road, the task was not as difficult as it could have been.
Mikel Arteta's side took the lead midway through the first half as Gabriel Martinelli stroked home a penalty after his shot struck Alex Iwobi's arm, with the spot-kick awarded after a video assistant referee check.
Eddie Nketiah nodded in following a corner to extend the hosts' lead on 31 minutes, before substitute Donny van de Beek side-footed in to make it 2-2 at the break.
Two second-half goals in three minutes ended the game as a contest. Both came from corners as Cedric Soares scored and Gabriel poked in.
The fifth place finish was two points behind Spurs. Their fans were able to celebrate a return to the Europa League after missing out on the top four.
They finished four points clear of the danger zone after their 13th away defeat.
The damage had been done in their previous two games, as they were beaten by Spurs and Newcastle, and they were unable to finish in the top four.
Granit Xhaka questioned his team-mates' character after the second loss, but they showed an appetite to make up for their performance at St James Park.
The home crowd booed Dele Alli on his first start for the Toffees, but that didn't deter Odegaard from being the key orchestrator and blocking two attempts. Martinelli was denied by Asmir Begovic, who replaced Jordan Pickford.
The storm appeared to be over initially, but then Iwobi gave the hosts a penalty.
The referee missed the shot, but after checking on the VAR monitor, he reversed his decision.
The goal was Martinelli's sixth of the season, and he had a hand in the second too, flicking on a corner, which Alli had failed to clear.
The two-goal lead allowed the hosts to settle, though there were chances for Demarai Gray before Rob Holding allowed the ball to run across him for Van de Beek's first goal since his loan move from Manchester United.
Two goals in three minutes put the match out of reach, and a second-half comeback akin to their heroics against Crystal Palace on Thursday never looked likely.
Despite the disappointment of missing out on the top four, the season can still be seen as progress for Arteta's young side.
The team extended their stay in the league on Thursday, which will be good news for the fans.
Since August, they have only won one away from Goodison Park, and a much-changed side showed a glimpse of attacking threat.
Richarlison and Pickford missed out due to injury. Tom Davies had a knee problem and made his first start since November.
For much of the first half, the side had to retreat to a back five, and although Gray drew a save from Ramsdale and fired over, they could not deal with the superior movement and energy of the team.
Three goals from corners were the most disappointing part of the game, with Michael Keane, Mason Holgate and Jarrad Branthwaite making up the back three.
The visiting supporters were in good spirits until the visitors took a three-goal lead, but they probably wouldn't have celebrated if they had stayed up.
When they heard that Manchester City had taken a 3-2 lead against Villa, they were still happy.
Formation 4-2-3-1.
17Cédric Soares16Holding6Gabriel20Tavares7Saka8Ødegaard35MartinelliFormation 4-5-1
17Iwobi4Holgate5Keane32Branthwaite2Kenny11Gray26Davies16Doucouré36AlliThe match ends with a single goal.
The second half ends with a score of 1.
The attempt was saved. Nicolas Pépé shot from outside the box is saved in the bottom left corner. Albert Sambi Lokonga assisted.
An attempt was missed. Price's shot from the right side of the box misses to the right. Dominic Calvert-Lewin assisted.
Gabriel Magalhães won a free kick in the defensive half.
Dominic Calvert-Lewin was the culprit.
The attempt was blocked. A shot from outside the box is blocked. Jonjoe Kenny assisted.
Foul by Nuno.
A free kick is won by Demarai Gray.
An attempt was missed. The right footed shot by Lacazette misses to the right. Gabriel Martinelli assisted.
An attempt was missed. Dominic Calvert-Lewin's shot from outside the box is too high. Anthony Gordon assisted.
Goal! The teams had 5 and 1 victories, respectively. The left footed shot from the centre of the box was taken by Martin. Nicolas Pépé assisted.
An attempt was missed. The shot from the left side of the box misses to the right. Martin was assisted by Degaard.
The foul was committed by Alexandre Lacazette.
The defensive half is where Michael Keane won the free kick.
There is a substitution for the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team from the team Nicolas Pépé is the new person.
There is a substitution forEverton. Tom Davies was replaced by Isaac Price.
The corner is at the football stadium. Conceded by Asmir.
The corner is at the football stadium. Conceded by Asmir.
The attempt was saved. Albert Sambi Lokonga's shot from the left side of the box is saved in the bottom left corner. Gabriel Martinelli assisted.
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