Arsenal have recorded a 3-1 victory over Everton at the Emirates Stadium but it was not enough to earn the Gunners a place in next season's Champions League.
Hector Bellerin, Alexis Sanchez and Aaron Ramsey all scored for Arsene Wenger's team, but Liverpool's 3-0 victory over Middlesbrough earned Jurgen Klopp's side fourth place in the Premier League table.
Romelu Lukaku netted a penalty for the Toffees but despite Laurent Koscielny's early dismissal, they could not take advantage.
Arsenal went into the game knowing that only a win would do against a team who sat two places below them in the table, and they should have taken the lead as early as the third minute when Bellerin set up Danny Welbeck, but his close-range effort went wide.
However, it did not take long for Bellerin to get on the scoresheet as he was able to score from inside the six-yard box when Welbeck missed his kick when he had the goal at his mercy.
It was one-way traffic in North London but on 14 minutes, Arsenal were reduced to 10 men when Koscielny was sent off for a crude challenge on Enner Valencia, with his dismissal also ruling him out of the FA Cup final.
Despite being at a man disadvantage, Arsenal doubled their lead soon afterwards when Welbeck atoned for his earlier miss by setting up Sanchez to score from just five yards out.
Everton were off the pace but as the first half progressed, they started to make inroads in the final third with Ross Barkley and Valencia having shots blocked inside the area, and Romelu Lukaku seeing a header tipped over the crossbar by Petr Cech.
Arsenal went into the break with the knowledge that Liverpool had opened the scoring against Middlesbrough, and the goals continued to fly in at Anfield at the start of the second half.
It effectively meant that Arsenal had failed in their attempts to finish in the top four, but they continued to try to make an impression against Everton with both Welbeck and Mesut Ozil being denied by Joel Robles.
However, just before the hour mark, Everton halved the deficit through Lukaku, who drilled a penalty past Cech after Rob Holding had handled inside the box.
With Liverpool cruising on Merseyside, the goal from the visitors was irrelevant in the grand scheme of things but it led to the match becoming more open as it headed into the closing quarter.
Ozil should have reestablished Arsenal's two-goal advantage when being played in by Aaron Ramsey but he was denied by Robles, while at the other end Valencia sent a shot wide of the post when he should have at least hit the target.
Alex Iwobi squandered two opportunities after being introduced as a substitute but in the final minute of added-on time, Ramsey sealed the win for the home side with a delightful curling effort into the corner after being teed up by Ozil.
