Niamh Fahey headed in the only goal of the fixture at Bromley FC to follow up last weekend’s 3-0 victory over London Bees for the Reds.
Liverpool made the better of the early running, with Melissa Lawley working the goalkeeper after moving in off the left-hand side to send in a shot.
And the Reds went even closer at the midway point of the first half.
Amy Rodgers created the opportunity with a through ball for Ashley Hodson to bear down on the Palace net, but Chloe Morgan denied the No.14.
Kirsty Linnett volleyed wide as the visitors continued to mount pressure, while Rachael Laws stopped an Eagles free-kick as the interval arrived with the score still goalless.