The fixture produced a relatively even contest for a good portion of the first half, with Leah Galton, Millie Turner and Ella Toone all coming close to finding an opener.
It would be the hosts to make the first mark, however, when Liverpool defender Jenna Clark headed home from close range at the back post, just minutes before the interval.
Liverpool managed to ask further questions following the break, buoyed by their goal late in the first half, but failed to double their advantage. United struggled to find a route to goal in the second half, coming closest in the final few minutes through substitute Melvine Malard's header, but otherwise found a tough Merseyside defence difficult to break down.