Is "by..." an adverbial phrase?

This is certainly an adverbial, explaining the condition fulfilled that enabled 'her' to open the door [and hence leave the room]. It modifies the whole matrix sentence (arguably only the first clause, with the compound predicate being better rephrased).

Some might classify this as an adverbial (nowadays an adverbial clause) of manner, though it really should be classed as facilitation / means (explaining how the opening was achieved, not just how it happened):

After seeing the expression of manner by means of adverbs [adverbials] [eg Tamara sings like a professional], let's move to deal with the expression of means and instrument. Means is expressed by a phrase [or clause] introduced by by:

• Tom Cruise always travels by plane.
• Beckham managed to sell the house by advertising it in a newspaper.

(On the other hand, instrument is introduced by a phrase introduced by with [or a clause introduced by using]:

• Victoria hit Ana with her bag.)

[Oposinet: The expression of manner, means and instrument; modified]