After Kane broke the deadlock with his 17th club goal this season, midfielder Oliver Skipp wasted a great chance to make it 2-0 as Tottenham eased off with some alarm.
Kane’s goal was well worth the wait, the 29-year-old exchanging passes in a congested area with Ryan Sessegnon before sending a right-footed shot into the opposite corner.
“I try not to think about it. Sometimes you can think too much,” Kane said as Greaves neared the landmark.
“I feel in good form and I feel fit. My mindset is to help the team and hopefully the goals will come.”
Premier League Leicester City, who won the cup in 2021, also reached the fourth round with a 1-0 win at Gillingham, who are bottom of the Football League.