The effect of an object's gravity propagates throughout the Universe; it just becomes significantly weaker with distance (exponentially, in fact). Newton's law of universal gravitation is:
Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.
So the spaceship is never actually out of the Moon's gravitational influence (as neither are we on Earth, hence the tides of the oceans), it's just that for much of its journey, Earth's gravity is much stronger. As the spaceship gets closer to the Moon, the pull of Earth's gravity weakens, until the Moon has more attractive force than the Earth. This happens gradually, not suddenly.