The Pilgrims led through Gavin Allott, only to be pegged back by Kayode Odejayi and Rowan Liburd before a Jay Rollins brace handed United a 3-2 half-time lead.
Brad Abbott and Lewis Gibbens were on target after the break in between two more Liburd efforts as the Pilgrims held on for a memorable three-point haul at Nethermoor Park.