Kane struck in the 77th minute at Wembley Stadium, three minutes after global teammate Kieran Tripper picked up where he left off at the World Cup with a curling free kick to put his side ahead for the second time.
Substitute Aaron Ramsey nearly equalised in the last minute but his strike landed on the roof of the net meaning new Arsenal boss Unai Emery has suffered defeat in his first two games since taking over from Arsene Wenger. Not think: "OK, we can win or we can draw".
"We know this team are probably going to take possession off us, and we must be ready to work all together off the ball, and show solidarity and great effort", he said. "It is three deserved points".
Tottenham Hotspur play their first home fixture of the 2018-19 Premier League campaign on Saturday, with newly promoted Fulham the visitors to Wembley.More news: Beetle preserved in amber reveals first insect-pollinated plant
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After the game Kane jokingly posted on social media, saying: 'Wish every game could be in August!'
Maurizio Sarri kicked-off his reign as Chelsea manager with an impressive 3-0 victory over Huddersfield on the opening weekend of the new season but the Italian will face a much stiffer test this evening.
A spectacular own goal from Matt Doherty and a deflected James Maddison strike put the Foxes in command. We want to challenge the big teams.
In stark contrast confidence at Bournemouth is sky high with the Premier League's longest-serving manager, Eddie Howe, leading his troops to maximum points from their opening two games. Two goals in six minutes turned the game on its head, though, as Callum Wilson and Steve Cook made it two wins from two for the Cherries.
The Brazilian, who should have been sent off in the first half for a wild kick out at Victor Camarasa and did not complete a single pass during the first 45 minutes, saw his nightmare afternoon ended with a poor penalty kept out by Neil Etheridge.
Tottenham beat Newcastle 2-1 last week.