The tie also saw Emily Ramsey forced off through injury, adding yet more casualties to the already depleted squad.
With injuries in mind, along with the visit of Manchester City on Sunday, Brian Sorensen fielded a number of youngsters, including a first senior appearance for 16-year-old Macy Settle.
The visitors started brightly and were ahead on five minutes through Sophie Roman Haug, with the Norwegian international directing Missy Bo-Kearns’ low corner past Ramsey.
The Blues stopper was the victor in the next dual minutes later though, holding a powerful Haug header from a rightfield cross.
The young Blues did enjoy periods of sustained possession during the opening 45 but failed to create any meaningful opportunities.
Sorensen looked to freshen things up at the break, with Clare Wheeler and birthday girl Aurora Galli making way for Heather Payne and Hanna Bennison, respectively.
However, it was Liverpool who continued to threaten and Ramsey had to be at her best to deny Jenna Clarke before Yana Daniels doubled the advantage for the visitors.
Yet again, it was another Kearns corner, with the Belgian bundling home from close range in a scramble that saw Ramsey pick up an injury and have to be replaced by Courtney Brosnan.
Payne looked bright and had the beating of her marker more than once. And the Irish star fashioned a decent chance for the Toffees when she managed to find Duggan, only for the striker to backheel straight at Rachel Laws.
As Everton pushed the game opened up and Brosnan picked up where Ramsey had left off to deny Rachel Kiernan, while at the other end Laws did well to stop an Emma Bissell effort.
But the Liverpool stopper was a mere spectator as Duggan unleashed from range only to see her strike hit the outside of the post before making the next chance count.
Sara Holmgaard collected on the edge of the area and found the veteran forward who rifled high into the Liverpool net for her second goal of this season’s competition.
The visit of Manchester City in the WSL on Sunday, 17 December, (kick off 1pm GMT) will round off the year for Everton, who will be thankful for the Christmas break if only to get one or two off the injury list and back on the field.