The league’s leading scorer this season took his tally to 39 his PL goals this year, surpassing Alan Shearer’s 36 in 1995.
Two Kane strikes from close range gave Spurs a 2-0 first-half lead.
Dele Alli put Spurs 3-0 up on 49 minutes with a curling shot. Two minutes later it was 4-0 when Son Heung-min fired in on the break.
Sofiane Boufal grabbed a goal back before Kane sealed his hat-trick with his 18th strike of the season.
Dusan Tadic came on to score a second for Southampton, who are 14th. Spurs lie in fifth position.
Spurs: 2 Jan v Swansea (A)
Southampton: 30 Dec v Man Utd (A)