Liverpool and City beat Arsenal to Champions League while Kane wins Golden Boot
Liverpool beat relegated Middlesbrough 3-0 at Anfield to secure a shot at the Champions League again next season as Arsenal finished outside the top four for the first time in 20 years.
The Reds knew victory would give them a place in the Champions League qualifiers in August, while Arsene Wenger's side had to hope other results would go their way as they tackled Everton.
After seeing an early penalty shout for Boro turned down, Liverpool finally settled the Anfield nerves when Dutchman Georginio Wijnaldum drilled them ahead just before half-time.
A 25-yard free-kick from Philippe Coutinho on 50 minutes and a third goal from Adam Lallana in the 56th minute secured the victory.
Arsenal, meanwhile, had to focus on beating Everton then hope results elsewhere also went their way.
The Gunners got off to a flying start when Hector Bellerin put them ahead on seven minutes, but then saw defender Laurent Koscielny shown a straight red card for a studs-up challenge on Enner Valencia which will see the France international suspended for the FA Cup final.
Arsenal doubled their lead in the 27th minute through a close-range effort from Alexis Sanchez.
Everton reduced the deficit with a penalty from Romelu Lukaku on 57 minutes following a handball by Nacho Monreal.
Midfielder Aaron Ramsey added a late third for the Gunners, who will now play in the Europa League next season - whether Wenger is in charge, though, remains to be seen.
Manchester City made sure of third place - and with it automatic entry into the Champions League group stage - after they swept past Watford 5-0 in what was Walter Mazzarri's last match in charge at Vicarage Road.
Skipper Vincent Kompany headed City in front after five minutes, with Sergio Aguero soon drilling in a second.
The Argentina forward made it 3-0 in the 36th minute, before Fernandinho and Gabriel Jesus rounded off victory as Italian coach Mazzarri saw his one season in charge end with a sixth consecutive defeat.
Departing Chelsea captain John Terry was given a guard of honour as he was substituted after 26 minutes - in honour of his shirt number - in the champions' 5-1 win over relegated Sunderland at Stamford Bridge.
The defender was handed his 492nd Premier League appearance for the Blues in their final home game, before being applauded off by team-mates and supporters alike.
Rock-bottom Sunderland took an unexpected early lead through Javier Manquillo but Willian soon had Chelsea level before play was halted for Terry's substitution.
Eden Hazard put the Blues in front just after the hour before Pedro added a third on 78 minutes. Substitute Michy Batshuayi scored twice in the final minutes to wrap up a record-breaking 30th Premier League for Antonio Conte's men.
England forward Harry Kane grabbed a hat-trick in Tottenham's 7-1 win at relegated Hull to secure the Golden Boot.
Kane - who netted four goals in the midweek 6-1 romp at Leicester - set second-placed Spurs on their way after 11 minutes and then swiftly doubled the lead.
Midfielder Dele Alli made it 3-0 just before half-time, with a deflected effort from Sam Clucas providing a consolation for the hosts after 66 minutes.
Spurs restored their advantage within three minutes through Victor Wanyama's header before Kane scored his 29th league goal of the season on 72 minutes.
Late strikes for Ben Davies and Toby Alderweireld completed the rout.
Manchester United's young side beat Crystal Palace 2-0.
With manager Jose Mourinho focusing on next week's Europa League final against Ajax, Josh Harrop marked his full United debut with the opening goal on 15 minutes - United's 100th different scorer in the Premier League's history - before Paul Pogba slotted in a second.
Burnley completed their Premier League season with a 2-1 home defeat to West Ham. Sam Vokes put the Clarets ahead but Sofiane Feghouli and Andre Ayew turned the game around.
Leicester and Bournemouth drew 1-1 as Junior Stanislas fired the Cherries ahead within the opening minute, but a 16th goal of the season for Jamie Vardy secured a point for the Foxes.
Fernando Llorente scored a late winner as Swansea came from behind to beat West Brom 2-1 at the Liberty Stadium. Jordan Ayew previously cancelled out Jonny Evans' opener.
Stoke beat Southampton 1-0 at St Mary's Stadium with a header from Peter Crouch on the hour.