United will go into next week's game away to Everton — now managed by Casey Stoney's former assistant, Willie Kirk — knowing that victory will clinch progress to the next round.
The Reds made the perfect start with just over 30 seconds on the clock. Green, fresh from being named FA Women's Championship Player of the Month for November, latched onto a pass from Jess Sigsworth and coolly fired home her 10th goal of the season.
Lauren James then twice went close to extending the lead, first crashing a curling effort against the inside of a post from the edge of the box before later seeing her powerful drive tipped onto the bar by Durham goalkeeper Megan Borthwick.