Learn German: Ich bin gestern sehr früh eingeschlafen. - I fell asleep very early yesterday.

In German, the Perfect tense usually uses haben, but verbs that describe a change of state or a change of location use sein (which is why we see bin here). Einschlafen (to fall asleep) is a classic change-of-state verb because you are changing from being awake to being asleep. If you were just saying that you slept, you would use haben (Ich habe geschlafen).

How is the past participle eingeschlafen formed?

The base verb is einschlafen, which is a separable verb. To make the past participle, the standard -ge- is placed right between the separable prefix ein- and the main verb schlafen. Because schlafen is an irregular strong verb, it keeps its -en ending instead of changing to a -t at the end.

Could I say Ich bin sehr früh gestern eingeschlafen instead?

It is grammatically understandable but sounds unnatural. When you have multiple time expressions in a German sentence, the rule of thumb is to go from the most general time to the most specific time. Therefore, the broader day (gestern) naturally comes before the more specific time of day (sehr früh).

What does the prefix ein- add to the word schlafen here?