Holbeach United booked a quarter-final place – despite missing eight players.
They hit Division One side Buckingham Town for six to set up a trip to Harrowby United in the next round.
Mitch Griffiths (2), Joe Smith (2), Cameron Gow and George Zuerner were on target for the Tigers.
Boss Seb Hayes said: “We dismantled them and it was probably our best display of the season.
“We had seven players out then Ollie Medwynter went back to Peterborough Sports on Thursday night.
“So I didn’t expect us to play like that from start to finish.
I remember going to Wellingborough last March when they beat us 3-0 so we can’t take them lightly.Seb Hayes
“We imposed our game on them and we were focused on making sure we started well.
“Our preparation for this game was awful as injuries and suspensions finally caught up with us.
“We’re trying to blood our younger players and now we’re reaping the benefits of the work by our management teams with the under-18s and reserves.
“Peterborough Sports did us a favour by letting us have Gow on dual registration. He’s only 18 but he has got pedigree from being at Peterborough United.
“We played Smith up front and he was outstanding as well. Every week he gets better.
“He is frustrated because he wants to play every game but he earned this chance.
“Now we can look forward to a quarter-final as well as the Lincolnshire FA Senior Trophy semi-final at Bourne Town.
“We’ve made no secret that we want to do well in all of the cup competitions but we also need to catch up on our league games.”
Holbeach will make the trip to Wellingborough Town on Tuesday as Luke Avis, Spencer Tinkler and Matt Warfield are back in the squad.
Stacy Cartwright, Charley Sanders and Jordan Keeble are still ruled out.
Hayes added: “I remember going to Wellingborough last March when they beat us 3-0 so we can’t take them lightly.
“I’ll be expecting another tough game. We can’t produce anything less than 100 per cent.
“We can’t under-estimate any team. If they click, we know Wellingborough can get three points in any game.”