Deeping Rangers booked their place in the preliminary round in a tight finish after a comfortable start.
The Clarets face a trip to last season’s Southern League Division One Central play-off finalists Barton Rovers on August 19.
Goals by Dan Schiavi, Jonny Clay and Brad Hockin earned Saturday’s victory with the bonus of £1,500 prize money.
Deeping led 2-0 at half-time but ex-Bourne Town striker Zak Munton and former Rangers forward Jezz Goldson-Williams reduced the arrears either side of a debut goal by substitute Brad Hockin.
Boss Michael Goode said: “We got through and the result is the most important thing.
“We totally dominated the first half and played some good football.
“We went 2-0 up and it really should have been quite a lot more.
“Jason Kilbride filled in for Scott Mooney up front and did a good job until he went off with a hamstring injury.
“We didn’t have a ruthless streak around the box when we created a number of chances.
“We were satisfied at half-time as we looked solid. We gave nothing away.
“We came out in the second half and our passing wasn’t quite right.
“We should have defended better but they made it 2-1 and we lost a little bit of focus.
“Hockin came on and got a goal to make it look pretty comfortable again.
“Richard Stainsby should have gathered the ball for their second goal but that made it nervy for the last few minutes.
“It was never going to be an easy draw away from home to a team from the same league.
“We’re happy to get the win without having played a league game this season which means we are not quite into our stride.
“It’s a tough draw in the next round but we are more than capable of getting the right result.”
Mooney will be back for the first game of the season in the ChromaSport & Trophies United Counties League’s Premier Division on Tuesday night at home to Wisbech Town.
Goode added: “We will have to play more consistently than we did on Saturday.
“If we can mirror the first-half performance then we’ve got a chance.
“But if we play like we did in the second half then it will be a difficult evening.
“Wisbech look strong and it should be a good game.
“Neither club will judge the whole season on one game at this stage.”