The striker has now scored 36 times in the English top flight in 2017 - the same number Newcastle United legend Shearer managed in 1995.
Kane scored a first-half penalty against Burnley before finding the back of the net twice after the break to achieve the feat.
The England international - who started the game on 33 goals - moved above Thierry Henry (34 in 2004) and Robin van Persie (35 in 2011) en route to levelling up with Shearer.
Kane, 24, will have the chance to set the all-time record when Tottenham take on Southampton in the league on Boxing Day.
7 - Harry Kane has now scored seven hat-tricks for Tottenham Hotspur in 2017 - five in the Premier League, one in the FA Cup and one in the Champions League. Hotshot. pic.twitter.com/a6DD3FnxgG— OptaJoe (@OptaJoe) December 23, 2017
46 – Only Edinson Cavani (50) and Lionel Messi (50) have scored more club goals in Europe’s big five leagues (all comps) in 2017 than Harry Kane (46). Elite. pic.twitter.com/rn7s3udNUY— OptaJoe (@OptaJoe) December 23, 2017
Only PSG hitman Edinson Cavani and Barcelona superstar Lionel Messi (50 each) have scored more club goals in Europe's top five leagues in all competitions in 2017 than Kane, who has 46.
Kane's hat-trick helped inspire Tottenham to a big victory over high-flying Burnley on Saturday with Mauricio Pochettino's men leapfrogging rivals Arsenal into fifth in the table in the process.