Roberto Firmino provided a stunning assist for Mohamed Salah's goal and was named man-of-the-match despite only coming on eight minutes before half-time.
Newly-promoted Norwich stunned Manchester City with a 3-2 win at Carrow Road on Saturday.
Mohamed Salah brushed aside his supposed falling out with Liverpool teammate Sadio Mane by posting a brilliant video.
Raheem Sterling enjoyed his best goal-scoring season last year as he netted 25 times to help City win the domestic treble of Premier League, FA Cup and League Cup.
A share of the spoils sees Tottenham's long wait for an away win in the Premier League dating back to January continue.
Brazil forward Richarlison scored the winner with 10 minutes left at Goodison Park to hand the Toffees their second win of the Premier League season.
Manchester United's season-opening 4-0 demolition of Chelsea is already a distant memory after Saturday's 1-1 draw against Southampton showed the flaws in Ole Gunnar Solskjaer's side.
Liverpool captain Jordan Henderson insists Sadio Mane's touchline tirade at Burnley showed his side's desire to win the Premier League.
Liverpool have now won a club record 13 straight Premier League games stretching back to last season when they missed out on the title to City despite amassing 97 points.
With Premier League leaders Liverpool off to a perfect start with three successive victories, City are already under pressure to keep pace.
Liverpool extended their perfect start to the Premier League season as Mohamed Salah inspired an impressive 3-1 win over Arsenal.
Frank Lampard had gone three games without a win in all competitions since he took over as Chelsea manager.
Aston Villa finally got their campaign up and running on their return to the top flight after a three-year absence.
Manchester United executives were keen for a meeting with Twitter after their French World Cup winning midfielder Paul Pogba was the victim of racist abuse.
Paul Pogba was assailed on social media, becoming the third player in England to have received racist abuse in the past week due to missing a penalty.
Manchester United would have gone top of the Premier League table with a win, but instead they were left to reflect on an inconsistent performance.
Frank Lampard had been given a thunderous reception in his first game as Chelsea manager at Stamford Bridge.
Manchester City were denied a win when Gabriel Jesus saw a stoppage-time winner disallowed on review for a handball by Aymeric Laporte.