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On another compelling afternoon in England's top-flight, Arsenal dented Chelsea's increasingly fragile title pursuit with a 2-0 derby win.
Alvaro Morata's second-half brace was enough for Chelsea, who beat United in last year's final, to move through.
Kane's 26th-minute spot kick was awarded after a lengthy, but correct, VAR decision and means Tottenham remain on course to claim their first silverware since winning the League Cup against Chelsea in 2008.
Southampton held Chelsea to a goalless draw on Wednesday, debutant Saints' goalkeeper Angus Gunn proving a bright spark in an otherwise lacklustre Premier League encounter at Stamford Bridge.
With the first nine penalties all converted, Chelsea striker Tammy Abraham saw his effort blocked by the legs of Liverpool keeper Adrian.
The 32-year-old was released by West Ham United at the end of last season but handed a lifeline by Liverpool boss Juergen Klopp.
Leicester finished the first half on top but couldn't translate their possession into real chances.
The win was United's biggest over Chelsea in a top-flight match since the same scoreline in March 1965 under Matt Busby.
Arsenal became only the second team in Europa League history to win a tie after losing the first leg 3-1.
Liverpool have 76 points from 31 games, two more than champions Manchester City who have a game in hand.
Results 201-210
