The 31-year-old Bayern Munich forward, who was honored with a golden cap before the game and wore eye-catching golden boots, finally broke through Finland's determined defense with a spectacular right-footed shot in the 57th minute.
At times it appeared to be a battle between Kane and Finland's impressive keeper Lukas Hradecky on a wet London night with the stopper making several fine saves to deny the striker.
But a razor-sharp Kane beat Hradecky again in the 76th minute, sweeping in another right-footed effort to take his England tally to a record-extending 68.
Kane, who also had a first-half header ruled out for offside, was given a standing ovation when he was substituted in the 80th minute.
England's win was their second under interim manager Lee Carsley after the away victory over Ireland on Saturday and puts them joint top of their Nations League group with Greece.