It was a third victory in four Premier League games for Spurs since Jose Mourinho succeeded Mauricio Pochettino last month, while they have scored 12 goals in that period.
Spurs scored twice in the first nine minutes to effectively end the match as a contest. Kane grabbed the first goal in the fourth minute when he lashed a shot past Nick Pope in the Clarets goal and into the top corner of the net.
Spurs doubled their lead five minutes later when Lucas Moura tapped the ball home after Dele Alli had got to the rebound first when Pope had saved Son Heung-min’s initial shot.
Sean Dyche’s side, who had lost 4-1 to Manchester City at Turf Moor on Tuesday, were in danger of being embarrassed and almost conceded another goal when Moussa Sissoko fired his low 15th-minute shot against a post.
Six minutes later Burnley went close to halving the deficit when an unmarked Robbie Brady headed Dwight McNeil’s cross against the underside of the bar before Chris Wood was unable to head the rebound home.
However, Spurs scored a third goal just past the half-hour mark with a scintillating Son goal, the South Korea international picking up the ball on the edge of his own penalty area before moving forward and slotting a 15-yard shot past Pope.
Spurs refused to rest on their laurels in the second period and Kane made it 4-0 in the 54th minute after an error by Matt Lowton.
Mourinho’s team finished matters off with a fifth goal in the 74th minute when Sissoko netted after playing a neat one-two with Kane.
The Spurs boss will be pleased that his side saw the match out and kept a clean sheet for the first time in 11 top-flight encounters.