While I agree with Eliezer that "emergence did it" is no more of an explanation than "god did it", I'm not sure that this is the meaning people are trying to convey when they say e.g. "intelligence is an emergent phenomenon". What the statement means to me is that intelligence is a computationally dense phenomenon. This doesn't rule out the possibility of understanding it from a lower level, but it does rule out _simplifying_ it to the point where it can be understood without performing large computations.
In my mind, the concepts of irreducible complexity and emergence are inherently linked through the concept of abstraction. An emergent phenomenon is a computationally dense phenomenon for which much of the computation can be abstracted away, at the cost of giving up a low-level understanding. In a sense, emergence is the antithesis of irreducible complexity. In another sense, emergence is a sparse chain of islands in a sea of irreducible complexity.