The Reds responded to a one-goal deficit in the FA Women's Championship clash with an improved second-half display that saw them establish a 2-1 lead in the wintry conditions.
Rachel Furness had brought her side level with a back-post finish, before substitute Ceri Holland marked her debut with what seemed to be a winner two minutes from time.
Charlton were awarded a last-gasp penalty, however, and converted it to grab a share of the spoils in dramatic fashion.
Hodson, who was introduced on the hour mark to play her first minutes of 2021 following a hamstring injury sustained in December, created both Liverpool strikes with crosses from the right flank.