That equalled the record set by Alan Shearer with Blackburn Rovers in 1995.
It was also the England international's seventh treble of the year, and took his tally to 53 goals in all competitions, as Spurs ran out 3-0 winners at Turf Moor.
Kane has now equalled Newcastle United legend Shearer's 22-year record and still has one game, at home to Southampton on Boxing Day, to make another record his own.More news: These parents got their child's teachers a hilarious personal gift
"It's a nice present that's for sure".
Speaking to BBC Sport, Pochettino said: "It was so important to us to perform how we did, to recover our feeling and level. It's a great way to finish before Christmas".
"I was a little bit frustrated we have not scored more - there was a massive chance for [Heung-Min] Son, for [Moussa] Sissoko, I feel sorry for them, but they need to keep going and the next games they are going to score, no problem".
There was a controversial moment in the fourth minute when Dele Alli was booked for a rash challenge on Burnley's Charlie Taylor, which might have resulted in a red card.More news: Apple slows some older iPhones because of flagging batteries
Kane nearly had a second goal a few minutes later when he rolled a shot narrowly wide from an opening created by Son Heung-min, but Moussa Sissoko was guilty of squandering a glorious chance to put Spurs further ahead midway through the first half.
"I thought he played great", said Kane of Alli.
"You always want to finish a game off away from home".
Deep into the first half, and with Burnley trailing 1-0 thanks to a penalty from Harry Kane, Wood looked to pounce on a loose back pass from Tottenham defender Serge Aurier, before keeper Lloris darted off his line.More news: Facebook challenges music streaming giants with Universal Music deal
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