England started brightly and scored after just five minutes when Keane fired the ball in after a goalmouth scramble.
They continued to dominate the early exchanges with Raheem Sterling pulling the strings in a free role in midfield. The Liverpool man missed an excellent chance on 30 minutes, heading over the bar from close-range after good work from Thomas Ince.
Their roles were reversed minutes later when Sterling raced down the right and played the ball into the box, but Ince fired over the bar from a threatening position.
A poor touch from James Ward-Prowse then allowed the goalkeeper to collect the ball when England were on the counter-attack before Ince fired wide from the edge of the box, courtesy of a deflection.
Then, just before the break, England broke the San Marino defence once more. Andrea Manzaroli fouled Sterling inside the box and the goalkeeper, perhaps harshly, was shown a red card. Kane stepped up and dispatched the spot kick to make it 2-0.