That should have given the Reds a comfortable half-time cushion but, after conceding a late corner, a poor touch in the box allowed Lauren Thomas to fire home a loose ball and reduce the deficit.
Ten minutes after the interval, though, the two-goal lead was restored as Olsson completed her hat-trick.
Good movement saw the forward meet a corner from Holland at the near post, with the flicked header too good for those on the line.
The Swede was withdrawn shortly after her third but the goals continued for Liverpool.
Enderby nipped past a defender on the edge of the box and fired a low shot into the corner with her left foot to make things safe on 63 minutes.
That completed the scoring but there were still real positives for the Reds in the remainder of the game.
Loan signing Emilia Szymczak gained her first minutes for the club, while Alejandra Bernabe and Lucy Parry both left the bench to return from long-term injuries.
The draw for the last eight will take place on Tuesday.
Team
Liverpool: Kirby, Fisk (Bernabe, 83), Clark, Evans (Szymczak, 69), Woodham (Parry, 83), Nagano, Maclean (Lundgaard, 69), Kapocs, Holland, Olsson (Kiernan, 62), Enderby.
Unused subs: Laws, Borggrafe, Bonner, Silcock.
Next up
The upcoming round of international fixtures means Liverpool return to action on Saturday December 6.
They travel to Emirates Stadium to take on Arsenal in a 12pm GMT kick-off in the Barclays Women’s Super League.
Away tickets can be purchased here.
