The England worldwide twice netted from close range- converting passes from Son Heung-min and Eric Dier- to take his Premier League tally to 98, one more than Teddy Sheringham scored for Spurs.
Both of Kane's strikes came in the second half, with the first coming when he got a toe on Son Heung-min's cross-cum-shot in the 47th minute.
The South Korean was again involved in the fourth, playing through to Alli, who flicked on to Christian Eriksen and the Dane finished in emphatic style.
England striker Kane has been talked up as someone who could break Jimmy Greaves' all-time Tottenham record of 266 goals and Alan Shearer's all-time Premier League record of 260.
Tottenham, on the other hand, thrashed Everton 4-0 at the Wembley Stadium on Saturday, moving to 44 points in the league table.
According to the Daily Mail, there are 34 Premier League players who are earning more than the 24-year-old, including West Ham's Javier Hernandez (£115,000 per week), Crystal Palace's Christian Benteke (£120,000 per week) and Liverpool's Daniel Sturridge (£150,000).
The Toffees opted for the long ball on numerous occasions as they looked to use the height of new signing Cenk Tosun.
Mauricio Pochettino: "He [Harry Kane] can do everything because of his mentality and how he is as a professional. He is always thinking to try to improve, improve, improve, score goals and improve for the team".
It was not necessarily an incredible performance from Kane, but to grab two goals despite not performing at his best shows just how clinical and important the Spurs talisman is.
Serge Aurier was played into space on the right and his low cross into the six-yard box was tapped home by Son.
The England striker has four goals in his last two meetings with the Blues and the league's top scorer will fancy his chances to add to his already impressive 18 goals on Saturday evening.
Kane's first goal, which put Spurs 2-0 up, appeared to be offside but Allardyce refused to make excuses for his players. I saw some good signs of the players last week against Liverpool and I saw some of the worst signs in the second half today which is massively worrying.