This example is interesting, because aside from interpreting eingeschlafen war as pluperfect of einschlafen it can also be interpreted as the past form of the same verb’s stative passive. A similar discussion can be made if comparing einschlief to eingeschlafen ist — which could be either stative passive or perfect tense.
Regardless of how you choose to interpret eingeschlafen war, however, it clearly references falling asleep as something that has basically already happened or started happening at the (past) point in time when the phone rang — this is the implication of the order of tenses. Einschlief, however, is more likely to mean that the phone rang at the beginning of his process of falling asleep — or maybe even just before it.
Additional discussion on the possibility of eingeschlafen ist
In general, the German perfect and preterite forms are freely exchangeable. Trying this out on your example sentence doesn’t work as well for me, though. This is because the perfect and stative passive forms are identical. Consider the following example:
Er ist eingeschlafen.
Without any additional details it is not clear whether the speaker is talking about the fact the he has fallen asleep at some point in the past or whether the speaker just intends to state his state of being asleep. It would need further marker words to distinguish between the two. For example:
Er ist gerade eingeschlafen.
This additional word shifts my perception of the sentence from both are possible to perfect tense.
Applying this discussion to the perfect version of your example sentence:
Als er gerade eben eingeschlafen ist, klingelte das Telefon.
Both versions are strong here. Gerade eben signalises a point in time which is more likely to mean perfect, while the conjunction als makes me more inclined to assume stative passive. Contrary to that, einschlief and eingeschlafen war are pretty clear-cut cases.