United took an early lead through a Rhian Cleverly own goal before Vilde Boe Risa doubled their advantage after the break.
Lewes kept pushing and Emily Kraft pulled a goal back with a fine lifted finish over United keeper Mary Earps.
But the visitors sealed their progress late on when substitute Nikita Parris prodded home a third.
It had looked as if United might run away with it after Lewes defender Cleverly tucked an Alessia Russo effort past her own goalkeeper to put the visitors ahead after only eight minutes.
But the hosts kept up a high work-rate and came close to an equalizer, with Natalie Johnson’s close-range effort blocked by a last-ditch Maria Thorisdottir tackle.
The Women’s Super League side took control after the break, with Sophie Whitehouse twice called into action to deny Martha Thomas, before Boe Risa nodded home a pinpoint Ona Batlle delivery to double United’s advantage.
Kraft took advantage of a poor Earps clearance in the 73rd minute to lift the ball over the England shot-stopper and give the hosts a glimmer of hope.
But United put the game to bed in the 89th minute, when Parris charged through the Lewes defense before poking an effort into the far corner from a tight angle.
Writing by Fany Olumoye; Editing by Julian Osamoto