With 20 minutes to go, Galton found Toone in the area but our no.7 shot wide and, shortly after, Liverpool were awarded another chance from the penalty spot.
Millie Turner brought down Jessica Clarke and substitute Babajide put the Merseysiders ahead.
Furness further extended the hosts’ lead eight minutes before the end when she headed in from a corner, leaving Casey Stoney’s side with work to do in Group C, where only the winner is guaranteed progress to the quarter-finals.
United: Ramsey; A.Turner (c), McManus, M.Turner, Smith; Ladd (Staniforth 46), Groenen (Zelem 61); Russo (Hanson 46), Toone, Heath (Galton 46); Ross (Press 61).
Subs not used: Earps.
Goal: McManus 28
Liverpool: Foster; Roberts, Robe, Fahey, Hinds; Furness, Bailey (Moore 71); Lawley (Kearns 81), Linnett, Clarke (Parry 90); Thestrup (Babajide 46)
Goals: Furness 33, 82, Babajide 72
Subs not used: Laws, Heeps, Ross, Parry
Referee: Rebecca Welch