The victory saw Spurs return to winning ways after their North London derby defeat by Arsenal and Antonio Conte's side sit three points off provisional table toppers Manchester City while Brighton dropped to sixth.
The win was also significant for Spurs as they paid tribute to their hugely popular fitness coach Gian Piero Ventrone who died earlier this week aged 61, with fellow Italian Conte visibly emotional during the minute's applause before kickoff.
"It's been a difficult week to say the least and it was nice to get the win," Kane told Sky Sports while his team mate Ryan Sessegnon said they "had to perform today".
"Obviously we wanted to win because of him and thankfully we did that," Sessegnon said.
"It's a difficult place to come, you saw their performance at Liverpool (in last week's 3-3 draw) and it was good for us to come here, keep a clean sheet and get a win. We had to dig deep."
Spurs took the lead in the 22nd minute when Son Heung-min's cross to the near post was deftly turned home by a glancing header from Kane, who was falling backwards to make contact with the ball to score his eighth league goal of the season.
Spurs coach Conte chose to break the norm and play a three-man midfield, with former Brighton man Yves Bissouma deployed in the middle of the park for the visitors.
But they sat back after the goal, happy to hit Brighton on the counter-attack.
Brighton responded well to going down with sustained pressure on their opponents, creating chances that tested Hugo Lloris in goal while Lewis Dunk and Solly March came close to scoring an equaliser in the first half.
Brighton looked keen to impress in new coach Roberto De Zerbi's first home game and continued to threaten after the break while Spurs also had their chances, with both teams defending well to stop promising attacks in their tracks.
But Spurs held on, becoming the first team to win at Brighton since their own victory in March last season.
"I must admit the result today is not fair. After the first 15 minutes there was only one team on the pitch and that team was Brighton," De Zerbi told BBC Sport."I don't believe in lucky or unlucky. The players were fantastic today, but when one team played like that today it's a good message for the league for the future."