Argentine midfielder Andres Gurrieri scored a superb 30th-minute winner, firing home from long range after Craig Reid had cancelled out Paris Cowan-Hall's early opener at Home Park.
Pilgrims boss Fletcher, whose side are now in Saturday's draw for the southern section quarter-finals, said: "I am really pleased. We had a really young side out, a game plan, stuck to it and got the early goal.
"It was pleasing the way the lads stuck at it because Aldershot are a big physical side, with a lot of tall players in there and we didn't have too many. We only had a couple of players over six foot."