Indirectly. The actual spike of pressure that ejects the magma comes from the gasses dissolved therein. When the magma moves up, the pressure drops and the gasses become oversaturated and thus prone to violent release.
The gas bubbles result form when pressure is reduced, like you mention. So there's actually less pressure near the surface, it's just more erratically applied. The pressure of the earth's subsurface is proving the animating force.