Michael Keane scrambled the ball home for the Young Lions in the fifth minute before Tottenham striker Kane took over the scoring duties with a 45th-minute penalty and two close-range efforts in the second half to seal victory.
San Marino goalkeeper Andrea Manzaroli was sent off after conceding the penalty when he brought down Liverpool's Raheem Sterling in the box, with the home side bringing on substitute stopper Massimo Francioni at the expense of midfielder Lorenzo Liverani as a result.
The win was England's second in three games in qualifying Group One and moved them up into second place in the standings behind Finland with a game in hand.
England totally dominated the game and San Marino had no answer once they had been reduced to 10 men.
Keane's opening goal proved to be the starter to Kane's main course, the 20-year-old Manchester United defender reacting quickly to re-direct Sterling's low drive.
The early goal set the tone and the Young Lions started to dominate, with Sterling flashing a shot wide in the 17th minute and England probing with a series of corners around the half-hour mark.
Ravel Morrison's volleyed effort flew over the bar before Tom Ince shot over, but England did get the second goal in first-half stoppage time when Manzaroli hauled down Sterling, allowing Kane to step up and beat substitute keeper Francioni from the spot.
Francioni was kept busy after the break, denying West Ham's Morrison and a point-blank header from Kane in quick succession, but Kane did find a way through in the 66th minute when he followed up his parried shot from close range.
Kane grabbed his third in the 89th minute when he found the end of club team-mate Tom Carroll's cross from the right.