Kane pounced in the 27th and 33rd minutes, both from Kieran Trippier assists, and Dele Alli also struck twice for the visitors with goals either side of halftime as Watford were punished for slack defending.
A fourth win in a row pushed Tottenham into third place with 39 points from 19 matches, above Manchester City and Arsenal, although their north London rivals could regain third spot if they beat Crystal Palace later on Sunday.
For the second game in a row Tottenham struck four on the road to keep themselves firmly in the pack chasing runaway leaders Chelsea, who they host on Wednesday.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
In torrential New Year's Day rain at Vicarage Road Tottenham had already peppered the Watford goal with shots before Kane opened the scoring.
Danny Rose fired wastefully over, Christian Eriksen stung the fingers of keeper Huerelho Gomes while Alli rattled the crossbar with a fizzing shot.
Seconds after Alli's near miss Trippier threaded a clever pass into Kane who reacted sharply and slipped the ball past Gomes.
Trippier, deputising for the suspended Kyle Walker, was at it again six minutes later, whipping in a teasing cross which Kane prodded past Gomes with Watford's defenders dozing.
The England striker, who scored once and missed a penalty in the 4-1 victory over Southampton in midweek, has now bagged 59 goals in 100 Premier League games and 10 in his last 11.
Watford continued to cause problems for themselves and a poor clearance by former Spurs defender Younes Kaboul fell to Alli who punished him with a tidy finish.
Alli grabbed his second after a mistake by Sebastian Prodl before he and Kane were both substituted with manager Mauricio Pochettino having one eye on Wednesday's clash with Chelsea who are 10 points ahead of Tottenham and six ahead of Liverpool.
"The Chelsea match is massive, so we'll recover and go hard at it," Kane told Sky Sports. "We just need to go out there and beat them and stop their winning run."
Watford grabbed a scrappy consolation goal in stoppage time when Kaboul touched home from close range after Spurs keeper Hugo Lloris had made a fine save.
(Reporting by Martyn Herman, editing by Mitch Phillips)
