Oldham manager Harry Kewell was relieved after his League Two side triumphed 3-2 at National League South hopefuls Hampton and Richmond.
The visitors were hanging on towards the end at the Beveree Stadium, but made it through to round two thanks to strikes from Ben Garrity, Bobby Grant and skipper Danny Rowe.
Kewell said: “I’m really glad to get through the tie.
“It doesn’t matter whether you beat someone by five or six, or just the one goal, it’s the result that counts and it was a tough game.
“It was a typical, traditional FA Cup game, and that’s what we expected.
“Once we saw the way Hampton set up, we knew it wasn’t going to be an easy afternoon.
“I’m just really pleased with the application from all of my players, though.
“The way they’ve stepped up the way they had to was excellent.”
The visitors were 2-1 up at the break thanks to close-range goals from Garrity and Grant.
They came either side of a successful penalty-kick from Sam Deadfield.
A clinical finish from Rowe extended Oldham’s lead just after the restart, before Deadfield’s close-range effort halved the deficit.
Borough manager Gary McCann was proud of his players.
He said: “We gave it all we had, and I couldn’t ask for any more as a manager.
“We caused Oldham all sorts of problems, but couldn’t quite grab ourselves an equaliser at the end.
“It was always going to be tough against Football League opposition, but overall we gave a great account of ourselves and the players should leave the ground with their heads held high.
“We’ve shown that we can compete with these teams, but perhaps it just wasn’t our day today.”