Liverpool began the second half in dominant fashion and Everton needed two fine saves by Brosnan to deny Jenna Clark and Cornelia Kapocs.
Brosnan somehow thwarted Thomas from point-blank range and the Blues then gained a foothold in the second half, with Falk making a superb stop to tip Imma Gabarro’s low strike wide.
But there was nothing the Liverpool ’keeper could do to prevent Everton from halving their deficit in minute 72 as Galli’s first-time hit rebounded in off Falk via the foot of the post.
Hikaru Kitagawa saw a deflected attempt well saved by Falk as the visitors sought to level, but the introduction of Aurélie Csillag off the bench gave the Reds an injection of energy and attacking threat.
Indeed, it was the hosts who looked most likely to score again in the closing stages and Olsson saw a goal ruled out for handball, but Liverpool had already done enough to earn their spot in the last eight.
Team
Liverpool: Falk, Shimizu, Clark, Fisk, Bernabe, Nagano, O’Sullivan, Kapocs, Enderby (Olsson, 38), Holland, Thomas (Csillag, 81).
Unused subs: Laws, Kirby, Evans, Woodham, Bonner, McDonald, Trueman.
Next up
Liverpool now have a three-week wait until their next game, with Leicester City visiting St Helens in the Barclays Women’s Super League on Sunday March 15 (11.55am GMT kick-off) after the international break. Ticket information is available here.