The comparisons between Mary and Eve are many; the one you mention being one of them.

Both Eve and Mary had choices, and Mary's choice was "Yes" and obedience.

There's no short answer to your question; the theology of the subject is deep and voluminous.

Here's a good article:
http://www.davidmacd.com/catholic/mary_mediatrix_co-redemptrix.htm

In in we read
"The Bible clearly shows us that man was given free will. Eve had free will to say "no" to God and 'cause the fall. Mary was free to say no to Gabriel. She was given free will.

So Catholics believe that Mary's role at the annunciation was special, not shared by any other human in the history of the world.  Mary's participation in the birth of Jesus was not passive. She had a role to play. Gabriel, the Angel, (who was a real gentleman) asked if she was OK with being the mother of the Saviour. If she said "no," none of us know what would have happened. Perhaps God would have worked out salvation history another way. Perhaps it was God's last chance for us, we just don't know. But when we think about the immensity of Eve's "no" we get a pretty good idea. Mary's "Yes" was huge."

Check out that link.  It's a good introduction to Mary's "like Eveness".

Something that I came across while reading St. Irenaeus that is related.

CHURCH FATHERS: Against Heresies, III.22 (St. Irenaeus)

Excerpt.

Wherefore also Luke, commencing the genealogy with the Lord, carried it back to Adam, indicating that it was He who regenerated them into the Gospel of life, and not they Him. And thus also it was that the knot of Eve's disobedience was loosed by the obedience of Mary. For what the virgin Eve had bound fast through unbelief, this did the virgin Mary set free through faith.