Scott Mooney and Luke Avis struck in a two-minute spell before the break as Deeping Rangers claimed another higher-level scalp.
Josh Oyinsan put Southern League side Barton Rovers ahead after just seven minutes but the Clarets hit back.
Boss Michael Goode said: “It was a funny game really as it wasn’t entertainining but we got the job done for a great result.
“They were a big, physical side and we probably gave them too much respect early on.
“They must have thought it was going to be easy and we didn’t do ourselves justice.
“Richard Stainsby made a couple of brilliant saves so we could have easily been 3-0 down.
“We raised our spirits and produced decent passages of play going forward which made us realise they weren’t that good after all.
“We were the better footballing side and following a set-piece, Mooney finished well.
“Straight after that, Avis took his chance clinically and it was a real bonus going in 2-1 up at half-time because we didn’t deserve it.
“They banged it long in the second half and we felt pretty comfortable. We didn’t look like conceding a goal.
“If Mooney wasn’t so tired, he could have scored again when he got through twice.
“Michael Simpson made his full debut and showed his experience in those situations by winning tackles as an old-fashioned central midfielder to help us get through. On the day, we were effective.”
The Clarets started kicking down the slope against a strong cross wind with Barton Rovers intent on getting the job finished quickly.
Jack Mace sent in a deep cross where Oyinsan challenged keeper Richard Stainsby with the ball ricocheting off the forward and over the bar.
Barton took the lead on seven minutes as Nathan Olokanmi burst to the goalline and a pull-back found Oyinsan who applied the sidefoot finish past Stainsby.
Deeping started to get their game going as David Burton-Jones sent in a left-wing cross and Henry Dunn headed back for Scott Mooney who was quickly crowded out by David Longe-King and Jordan Odofin.
The clearance found Olokanmi on halfway and the pacy winger sped into the Deeping box where Jonny Clay was on hand to block away.
Yemi Adel’s curled cross found Oyinsan in the Deeping box where Clay and Charlie Ward challenged but he still managed to get a shot in that Stainsby saved well.
At the other end. Mooney twisted into some space with the shot being blocked away.
A loose ball from Dunn allowed home skipper Claudio Dias to release Adelani who advanced to the edge of the box to blast goalwards where Stainsby made a good save.
A rare error from Clay allowed Oyinsan free down the right, his cross saw Adelani shoot for Stainsby to make an even better save.
Rangers levelled on 42 minutes as Dan Flack’s deep free-kick was met at the far post by Charlie Ward’s header into the goal area and Mooney applied the finish.
Two minutes later, they took the lead. Dunn’s pass picked out Mooney whose touch allowed Scott Coupland space to pick out Avis in the box for a drive which saw the net bulge.
Deeping had the stronger wind at their backs for the second period and started out in their normal manner.
They were rarely threatened after Ebby Nelson-Addy’s pass over the Rangers backline allowed Olokanmi to run into the box where Ward’s pressure saw the shot at Stainsby.
Rangers were controlled and the game became more box to box.
A mis-kick by Odofin allowed Mooney to run at the home defence, only to scuff his shot under pressure from Longe-King.
Burton-Jones thrust forward from halfway into the home box and, despite pressure, managed to nudge the ball past advancing keeper Lewis Todd to see the ball cleared from the line by Ryan Haugh.
Rovers’ last chance to salvage the game saw Oyinsan onto a loose ball on the edge of the box, the shot was easily taken by Stainsby.
Rangers who will be looking for their next opponents when the draw is made on Monday.