Match ends, Burnley 1, Arsenal 3.
Pierre-Emerick Aubameyang scored twice as Arsenal beat Burnley at Turf Moor, but the Gunners failed to finish in the top four for the third successive season.
Unai Emery's side secured fifth place meaning they go straight into the Europa League group stage next season.
And they can still qualify for the Champions League if they beat Chelsea in the Europa League final on 29 May in Baku.
Aubameyang, who scored a hat-trick on Thursday as Arsenal beat Valencia in the Europa League semi-finals, finished the season with 22 league goals.
That saw him tie for the Premier League Golden Boot with Liverpool duo Sadio Mane and Mohamed Salah.
Ben Mee gifted Aubameyang his first, miscontrolling a pass from Jack Cork to allow the Gabon striker to run in and slot past Tom Heaton on 52 minutes.
He then lashed in a volley from close range 11 minutes later and should have had a hat-trick, but with the goal at his mercy, he poked wide.
Burnley, who had already secured a fourth straight season in the Premier League, pulled one back as Ashley Barnes met Johann Berg Gudmundsson's pass with a glancing header.
Arsenal's 19-year-old forward Eddie Nketiah added a third with the last play of the game in the 95th minute as he poked the ball through Heaton's legs.
No local bragging rights for Arsenal
To finish in the top four, Arsenal needed to beat Burnley and hope rivals Tottenham lost against Everton - while also achieving an eight-goal swing.
As it was, Spurs drew 2-2 with Everton to take the final Champions League place, a point ahead of Arsenal.
They have now finished below Tottenham in three consecutive league campaigns for the first time since a run of nine seasons from 1959-60 to 1967-1968.
That means there will not be a return of St Totteringham's Day, which was the point in each season when Arsenal fans marked the precise moment they guaranteed finishing above Spurs.
The Burnley win did at least snap a run of four games without a win in the league.
Emery made nine changes from the side that beat Valencia 4-2, with only Aubameyang and Nacho Monreal keeping their place.
Aubameyang threatened twice early on, shouldering the ball against the crossbar from a free-kick and seeing a shot saved by Heaton from a tight angle.
But Burnley grew into the game and dominated against Arsenal's defence of Shkodran Mustafi and Konstantinos Mavropanos.
An injury to Mavropanos in the 34th minute allowed Laurent Koscielny to come in and steady the ship.
Aubameyang proved the difference as he made it six goals in a week and took his tally to 31 for the season in all competitions.
"I happy we won, and I am sharing this [Golden Boot] trophy with two other guys I like," said Aubameyang.
"My team-mates knew about the Golden Boot, I said nothing. I did not want them to play and only focus on me. I am a team player. The team have had a great season but last month it has not been the best in the league."
A Burnley goal always seemed likely and, when it arrived, it was the 51st Arsenal had conceded in the league this season, matching last season's total as the most they have conceded in a Premier League campaign.
All eyes are now on the trip to Baku and a chance to win not only a trophy but get back into Europe's elite.
Following that, Emery will need funds in the summer if he is to shore up his leaky defence while bettering this season's tally of just seven away wins will also be crucial if a top four finish is to be achieved. There could also be more involvement for Nketiah who scored his first Premier League goal in a lively cameo.
"I am proud of the players, we deserve to be fighting for the top four. At the last moment we didn't take it, but we are closer to the other teams," said Emery. "Our first target in the league we can not take, but we have the Europa League to play for one title and this [Champions League] target.
Burnley's long season finally over
A 15th placed league finish is reward for a tough season that started on 26 July against Aberdeen in the Europa League qualifiers.
Their league form suffered and they sat in the relegation zone at Christmas and they never looked like reaching the heights that took them to seventh place last season.
But with safety secured at the end of last month, Sean Dyche could look forward to a long summer off and a full pre-season.
Not that his side played like their minds were on their upcoming holidays as they dominated for large parts of the game. Chris Wood went closest when his shot hit the inside of the post, while Matthew Lowton's shot from close range was blocked by Mustafi in the first-half.
They battled hard and, in a feisty encounter, five Clarets players ended up in Mike Dean's book.
Even after going behind, they showed character to pull one back and substitute Peter Crouch also went close at the end.
"I was pleased with today's performance but the cutting edge was lacking," said Dyche.
"I have been proud of the players all season. The way they have taken on the second-half of the season. They were under duress around Christmas, the noise outside the camp was people saying we were finished, but the way they have seen through that, concentrated and focused on the challenge in front has been magnificent."
Man of the match - Pierre-Emerick Aubameyang
Arsenal love the final day - the best stats
- Burnley have lost each of their last 10 meetings with Arsenal in all competitions
- Arsenal haven't lost on the final day of a Premier League season since the 2004-05 campaign at Birmingham. The Gunners have won their final Premier League game of the season 18 times - no side has done so more
- Arsenal have kept just one clean sheet away from home in the Premier League this season, their fewest in a single top-flight campaign since 1967-68 (also one)
- Eddie Nketiah is the first teenager to score a Premier League goal for Arsenal since Alex Iwobi netted against Watford in April 2016
- Aubameyang (31) is only the second Premier League player to score 30+ goals in all competitions this season, after Sergio Aguero
- Barnes has scored 12 Premier League goals this season, more than any other Burnley player in a single campaign in the competition
- 1HeatonBooked at 3mins
- 2LowtonBooked at 48mins
- 5TarkowskiBooked at 45mins
- 13HendrickSubstituted forBradyat 82'minutes
- 4CorkBooked at 75mins
- 31McNeilSubstituted forBerg Gudmundssonat 64'minutes
- 10BarnesBooked at 22mins
- 11WoodSubstituted forCrouchat 77'minutes
- 7Berg Gudmundsson
- 27MavropanosSubstituted forKoscielnyat 34'minutes
- 29GuendouziBooked at 52mins
- 59WillockSubstituted forNketiahat 62'minutes
- Mike Dean
- Shots on Target
Second Half ends, Burnley 1, Arsenal 3.
Goal! Burnley 1, Arsenal 3. Eddie Nketiah (Arsenal) left footed shot from a difficult angle on the left to the centre of the goal. Assisted by Alex Iwobi.
Attempt missed. Pierre-Emerick Aubameyang (Arsenal) right footed shot from the centre of the box is just a bit too high. Assisted by Alex Iwobi.
Attempt blocked. Henrikh Mkhitaryan (Arsenal) right footed shot from outside the box is blocked. Assisted by Mohamed Elneny.
Attempt missed. Ashley Barnes (Burnley) header from the centre of the box misses to the right. Assisted by Johann Gudmundsson with a cross.
Foul by Henrikh Mkhitaryan (Arsenal).
Ashley Westwood (Burnley) wins a free kick in the attacking half.
Foul by Nacho Monreal (Arsenal).
Johann Gudmundsson (Burnley) wins a free kick on the right wing.
Offside, Burnley. Ashley Westwood tries a through ball, but Johann Gudmundsson is caught offside.
Attempt blocked. Peter Crouch (Burnley) left footed shot from the centre of the box is blocked. Assisted by Ashley Barnes.
Substitution, Burnley. Robbie Brady replaces Jeff Hendrick.
Foul by Henrikh Mkhitaryan (Arsenal).
Jeff Hendrick (Burnley) wins a free kick in the defensive half.
Attempt blocked. Alex Iwobi (Arsenal) right footed shot from the left side of the box is blocked. Assisted by Mohamed Elneny with a cross.
Attempt blocked. Eddie Nketiah (Arsenal) right footed shot from the centre of the box is blocked. Assisted by Alex Iwobi.
Substitution, Burnley. Peter Crouch replaces Chris Wood.
Corner, Arsenal. Conceded by James Tarkowski.
Jack Cork (Burnley) is shown the yellow card for a bad foul.