Generally speaking, an eclipse doesn't necessarily have to be physically affecting the entire world, just a small segment of it for a story to carry out. Therefore, in my mind, a powerful being or event that creates an eclipse without literally destroying the entire world or moon by moving it near instantaneously in the atmosphere, would be either through illusion or overlapping dimensions.
Especially overlapping dimensions, consider the Shadow Realm from World of Warcraft - it is an exact replica of the normal Universe, however, Death Knights of the Lich king have the ability to pull, or project an object from that realm. So really, this can be answered in a lot of different ways.
Probably the easiest, as in, uses the least amount of energy, would be to just project an image of a moon into the sky that blocks out the sun. To explain multiple mythologies co-existing, the deities that reign over similar elements, objects, etc. aren't ruling over the physical objects themselves.
For instance, Egyptian Ra, where the Sun is still in the same place, however a given deity believes they have the ability to utilize its power.Belief is often the driving force in mythology, so while the deity believes this is the case, that makes it true. Of course, in reality, they kind of just control the effects of the thing they reign over, how it affects their region of influence.
Therefore, if something had "power over the moon", they're really just removing people's perception of the real one and replacing it with their own, and the effects that come with it. You can think of it like those Sci-Fi Nature domes in spaceships - it looks and generally feels like the moon, but in technicality it isn't. It could be a lot more complex than just a screen over the area, it could do with manipulating beings perception of their surrounding.
Does that make sense? Broadly speaking, abilities and powers like that in intertwining myth don't often have literal control over something, just the perception/idea of it that's physically believable enough that it's nearly indistinguishable.
EDIT: To add on even more, for specifics, projections could be by magic illusions, targeted psychological projection, "window" to a different realm/dimension, physical manifestations, and more, but the idea of projection opens a lot of flexibility and technicalities in a being's/event's scope of power.