Villumsen, who missed out on bronze by 1.8sec at the London Olympics four years ago, said she felt flat right from the start.
"Not real good - from the beginning, I was a bit tensed up. Usually you're better after 10 or 15km but today it just didn't happen."
The world and Commonwealth champion, Villumsen recorded a time of 44 minutes 54.71 seconds and 28.29sec behind Armstrong who won her third consecutive gold medal in the event.
The Kiwi was fifth through the first time check at the 10km mark, almost 18 seconds behind Armstrong and held that position at 19.7km.
She had targeted an Olympic medal to add to her successes in the event but ended almost 17sec behind Dutchwoman van der Breggen in bronze.
- AAP