The striker failed to score in all three of Tottenham's Premier League games in August, extending his dry run in that month to 12 matches for the club in total.
Kane got off the mark for this season, scoring the first and last of England's four second-half goals against the world's 190th-ranked team in Group F of European qualifying.
"Didn't like August anyway!" Kane, the top scorer for the past two seasons in the Premier League, wrote in a lighthearted comment on Twitter soon after the match.
England's build-up play was ponderous and only improved once Manchester United forward Marcus Rashford came on as a halftime substitute to inject some directness in the visitors' play.
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"Whenever you come away for games like this, it is never going to be easy," said Kane, who now has 10 goals in 20 international appearances. "We were trying to break them down, (but) they had 10 men behind the ball."
The English are unbeaten in qualifying with three games left, having won five and drawn two. They stayed two points clear of Slovakia, which beat Slovenia 1-0 in Trnava to boost its chances of at least securing a spot in the playoffs for the eight best second-place finishers.
SLOVAKIA 1, SLOVENIA 0
Adam Nemec scored the only goal, bundling in from close range in the 81st minute after a cross from the byline struck the underside of the cross bar and bounced into the path of the forward.
Slovakia heads to Wembley looking to avenge a painful loss to England in the first round of matches, when Adam Lallana scored in the fifth minute of stoppage time in a 1-0 win in Trnava.
LITHUANIA 0, SCOTLAND 3
Stuart Armstrong, Andrew Robertson and James McArthur were the scorers in Vilnius, ending Lithuania's unlikely hopes.
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