We are in escrow for a home in Los Angeles. Escrow closes on Oct. 13.
If we put in a 30 day notice with our landlord (vacating November 1) and fall out of escrow, are we legally bound to move out of the apartment? Or should we wait till escrow closes to put in our 30 day notice?

I would wait till you are in escrow, since a closing date is moved and set back on regular basis all the time.

What type of landlord is it and how good a relationship do you have?

If you are in a large impersonal complex, then I would wait until you are sure of your move out date. This may cost you extra, but it will cost you more if you need to move into a hotel because your house isn't ready.

If you are in a house or small complex and you are on good terms with the LL, then explain the circumstances to them and ask what they want.

Legally, you have to move the day that you say that you will in the notice. After all, your landlord may have rented the unit to someone else already, because you said you would be out of it. I'd wait until I was pretty sure escrow was going to close before putting in my notice, if I were you.