Kane scored the only goal of the game at Wembley when he sent a towering header past Petr Cech after rising above Laurent Koscielny in the 49th minute to meet a Ben Davies cross.
The strike was Kane’s seventh in seven north London derby appearances.
The result ensured Tottenham put clear distance between themselves and the Gunners and climbed above Liverpool into third spot in the Premier League table – just four points behind second-placed Manchester United.
Morgan, meanwhile, tried to taunt Kane before the match when he tweeted in reference to Pierre-Emerick Aubameyang’s excellent goalscoring record by tweeting: “Can’t wait to see the best striker in Europe tomorrow”.
Aubameyang was tagged in the tweet and Kane was copied, and the England international acknowledged him by quoting the original tweet and responding with a simple “Thanks”.
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Kane’s strike was his 23rd in the Premier League this term.
Eric Dier suggested after the game that his team-mate could have scored a hat-trick if he wanted.
He said: “We dominated the game and second half we countered really well, created lots of chances.
“It’s disappointing that we struggled in the last five minutes as the game should have been settled.
“This team has created the mentality where we can dig in if we have to.
“They had chances at the end but it could have been a much bigger win.”