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The first point is that not everyone would accept the presupposition of your question. Most obviously, theists wouldn't. According to many varieties of theism, the First Cause of the material world is not a material thing. Needless to say, not everyone agrees. But you can deny that there is a non-physical First Cause without denying that the very idea is incoherent.
There are homelier examples. On at least some views, the fact that something was absent can be a cause. Absences, however, aren't material objects. (In fairness, they aren't non-material objects either.) So the first point is that it isn't simply agreed by the parties to the dispute that only material objects are causally efficacious. We could also add that even among materialists, broadly understood, most would say that events rather than objects are what do the causing, but it's at least arguable that events are in space and time and so even if they aren't material objects, they're physical in a broader sense.
The second point is that there are different theories of causation, and on one important approach, causation should be understood in terms of counterfactuals. This way of thinking about causation goes back to David Hume, although he didn't develop the idea in any detail. The most important recent defender of a counterfactual analysis of causation was David Lewis, and we can illustrate the idea with a simple case. Suppose that if a certain switch were flipped, a light would turn on. And suppose that if the switch were not flipped, the light would not turn on. Then on this sort of view, flipping the switch causes the light to turn on. The more general idea is this: suppose C1, C2,... Cn are mutually exclusive jointly exhaustive possible events (that is, no two can occur at one, and together they exhaust the alternatives.) Likewise, suppose E1, E2,...En are mutually exclusive, jointly exhaustive possible events, all distinct from the C-events. Suppose that the following are all true:
If C1 occurred, E1 would occur.
If C2 occurred, E2 would occur
....
If Cn occurred, En would occur.
Then the E-family of possibilities depends causally on the C-family. But notice that this approach says nothing about whether causes are material or immaterial. True, as I've presented it, the story relies on the idea of something occurring. That at least invokes time. But there are ways to make the analysis general enough to include, for example, the possibility that the cause of the existence of the material world is the (timeless) fact that its existence is God's will. (Had it not been God's will that the universe exist, it would not have existed.)
Whether a counterfactual analysis of causation is the best way to go is a matter of debate. But the point is that there is a real debate here. You might find the discussion and references in this article from the Stanford Encyclopedia of Philosophy a good place to start.
https://plato.stanford.edu/entries/causation-metaphysics/