Mohamed Salah broke the Premier League scoring record for a 38-game season as Liverpool qualified for the Champions League on a busy final day of the season.
Salah's 32nd goal - taking him past Alan Shearer, Cristiano Ronaldo and Luis Suarez, a Dejan Lovren header, Dominic Solanke's first goal for the club and Andrew Robertson's goal secured victory.
They needed only a point to head off Chelsea and seal the fourth remaining spot in Europe's premier club competition next season.
Victory was not enough to take Liverpool up to third place, which was claimed by Tottenham with a stunning 5-4 win over Leicester in what may be their final game at Wembley before moving to their new stadium at White Hart Lane.
Mauricio Pochettino's side trailed three times, but a brace each for Erik Lamela and Harry Kane - the latter reached 30 goals for the league season as a result - along with an own goal from Christian Fuchs gave them victory.
Kane's England team-mate Jamie Vardy scored two of Leicester's four goals.
If Liverpool had lost to Brighton, they would not have missed out on the top four as a result of Chelsea's 3-0 loss at Newcastle.
Ayoze Perez scored twice after Dwight Gayle's opener to ensure Rafael Benitez's Newcastle finished 10th in the table, while Antonio Conte's side had to settle for fifth and a shot at the FA Cup in Saturday's final against Manchester United.
Swansea needed to win, Southampton lose to Man City and there be a 10-goal swing to avoid being relegated on the final day. Swansea took the lead against Stoke but fell to a 2-1 defeat and a place in the Championship next season.
An injury-time goal from Gabriel Jesus consigned Southampton to defeat but, more significantly, gave champions Manchester City the win they needed to end their remarkable season with a record 100 points.
Arsene Wenger's final game of his 22-season Arsenal career was a victorious one as Pierre-Emerick Aubameyang's first-half goal gave them a win at Huddersfield.
Both sets of fans applauded Wenger in the 22nd minute of the game and two planes flew overhead dragging messages of support for the Frenchman.
West Brom's stay in the Premier League ended with a 2-0 defeat at Crystal Palace, for whom Wilfried Zaha and Patrick van Aanholt scored.
In a game between two sides managed by a former boss of the other team, David Moyes' West Ham came out on top against Sam Allardyce's Everton, with Manuel Lanzini scoring twice in a 3-1 win.
Marcus Rashford scored the only goal in Manchester United's 1-0 home win over Watford, while Callum Wilson's 93rd-minute strike gave Bournemouth a 2-1 comeback win at Burnley.
In Scotland, Rangers and Hibernian shared 10 goals in a superb game at Easter Road.
Hibs took a 3-0 lead inside 22 minutes but Rangers levelled the score before the break. The away side looked on course for victory, leading 5-4 going in to the last five minutes but Jason Holt's sending-off left them vulnerable and Jamie Maclaren completed his hat-trick to rescue a point for Hibs.
Aberdeen became the first Scottish team to inflict a home defeat on Brendan Rodgers' Celtic as the Dons secured second place in the Premiership and limited Rangers to third.
Lee Erwin's early goal was enough to give Kilmarnock a 1-0 win over Hearts and seal fifth place with their highest Premiership points tally of 59.