City began the second half on the front foot but, five minutes after the restart, the Reds put together their best passage yet.
Ceri Holland won possession with a high press and prodded a pass in behind for Mia Enderby, whose attempted centre was deflected onto the near post.
But those on the Kop were lifted off their feet for real moments later.
The deadlock was broken in classy fashion, Grace Fisk spreading play out to the right wing and Holland in turn clipping a peach of a cross into the City box.
Arriving right on time at the back post was Kapocs, who bravely put her head in to nod the ball home.
It would prove to be a short-lived advantage, however, as City levelled with 64 on the clock.
Alex Greenwood’s delivery from the channel was glanced by Shaw towards goal. Borggrafe made the save but could not keep hold, allowing Beney to blast in the rebound.
City cranked up the pressure having got themselves on terms and Borggrafe was needed again 11 minutes from the end, palming a Shaw thump behind.
The away side produced a decisive goal on 86, though. A slick passing move infield from the right freed Fujino to rasp a shot in.
In the last act of the encounter, Bonner smashed in the loose ball when Holland’s free-kick strike was fumbled – but the offside flag cut short the Kop-end celebrations.
Team
Liverpool: Borggrafe, Fisk, Bonner, Clark, Evans, Nagano, Kerr (Roman H., 90+3), Höbinger (Maclean, 26), Holland, Enderby, Kapocs.
Unused subs: Kirby, Shimizu, Szymczak, Olsson, Lundgaard, Silcock.
Next up
Taylor’s team will now turn focus to the Subway Women’s League Cup in midweek, travelling to Durham in the group stage.
Kick-off at Maiden Castle is 7pm BST on Thursday (October 16).
