Kane received the ball from Pierre-Emile Hojbjerg, then turned and drilled a low shot into the corner from the edge of the area.
Steve Bruce's men are winless in eight games in all competitions, including exits from both domestic cups.
Chelsea midfielder Hakim Ziyech believes that his team could have scored more goals but is satisfied with the performance.
Liverpool trailed to Sander Berge's controversial first half penalty at Anfield, with the Reds frustrated that VAR gave the spot-kick despite Fabinho's foul appearing to take place outside the area.
The two teams had lost each of their first four league games this season, raising the stakes for their meeting at Bramall Lane.
Bamford headed home from Jack Harrison's left-wing cross and has now scored a goal in all three of Leeds' games on the team's return to the top division.
Englishman Chris Wilder has been in charge of the club since 2016, overseeing two promotions in three seasons.