Your question makes an unstated assumption: all three bodies are always in the same plane. In fact, if this were true, there would be a solar eclipse for every new moon. So your spatial imagination is working quite correctly, indeed!
However, the unstated assumption does not hold. The plane of the Earth orbit around the Sun (the Ecliptic) is not the same as the plane of the Moon orbit around the Earth. Now lets test your spatial imagination again: what is the intersection of the Ecliptic with the tilted plane of the Moon orbit? Right, a straight line, with the Earth as a common point to both planes. And if you ask, "what is the intersection of the Ecliptic with the Moon's orbit (almost an ellipsis)?" the answer is: just two points. These points are called nodes.
For complex reasons, these nodes are not stationary, but move in time within the Ecliptic, around the Earth. Now visualize again what happens when a node happens to have moved between Sun and Earth and the moon happens to be at this node in her orbit at the same time: right, a Solar Eclipse. Voilà!
The other node, when extending the line from Sun to Earth, is the one responsible for Lunar eclipses. Since both nodes are opposite and move slowly around the Earth, solar and lunar eclipses often precede or succeed each other within the two weeks it takes the Moon from one node to the other.