Recent away struggles for Spurs, coupled with Burnley's impressive form, suggested the hosts could claim another big scalp.
Spurs boss Mauricio Pochettino said afterwards: "First of all I have to congratulate Harry - it's the same record as Alan Shearer. Then the team - they were amazing. We needed this type of victory and we got the three points.
"It's difficult to find words to describe Harry Kane. Even during my time at Spurs I'm surprised by him. The hat-trick is nice for him but we could have scored more than three. I'm worried when you don't create chances though.
"We were really solid - it was a solid performance. We recovered our spirit in the way we played in the last two of three seasons.
On the first-half penalty, he added: "The penalty was a penalty. The crowd started to focus on Dele Alli [after he won the penalty]. He has a great personality and character and you can't intimidate him.
"We are now in a position we can fight for the top four which is important after the loss to Manchester City."