In the 21st minute Zigiotti Olme stepped forward and struck a superb long-range finish to make it 2–0 to United.
The goal was the 28-year-old's first for the club since joining last summer.
United were able to increase their lead to 3–0 in the first half, before the team added two more in the second half to make it 5–0.
Hanna Lundkvist also started the match for United, and Anna Sandberg was substituted with half an hour remaining. Ellen Wangerheim, who was signed by United earlier in the week, made her debut when she came on at the same time.
In Liverpool's FA Cup meeting with the London Bees, former Häcken player Alice Bergström scored her first goals for the club.
Bergström was brought on with just over 20 minutes remaining, and before the clock reached 90 minutes she had scored twice in two minutes, making it 6–0 to Liverpool.
Bergström was signed by the Merseyside club in early January.
National team goalkeeper Jennifer Falk, on loan from Häcken to Liverpool, played from the start. Cornelia Kapocs remained on the bench for the entire match.
Both Manchester United and Liverpool have qualified for the fifth round of the FA Cup with their victories.
