Mark Jones gave the Pilgrims the dream start at The Jakemans Stadium, but Ben Fairclough was quickly red-carded for a challenge on Matt Doughty.
James Lawrie equalised after the break and once Spencer Weir-Daley had been sent off for apparent violent conduct, United were up against it.
Jake Moult and Nicky Clee scored late goals to book Altrincham's place in round one and end the plucky efforts of the nine-man Pilgrims.