The Imps scored the only goal of the game in the final minute of normal time through Sean Raggett to create cup history, while at the same time becoming just the fourth team in 30 to leave this ground with victory.
A strong Burnley side struggled to break down their opponents across the 90 minutes, setting up a dramatic conclusion in Lancashire as the National League side produced one of the all-time great results in the competition's history.
That would prove to be the visitors' cleanest sight of goal in a tight first half, with Burnley edging things through a scuffed Andre Gray shot down the middle and Joey Barton's edge-of-the-box volley for Paul Farman to again easily keep out.
Burnley named a strong side for this match in the hope of keeping their own cup dream alive, making just a few changes from the draw against Chelsea here a few days ago, but they were struggling to create any clear-cut openings.
Burnley, whose wait to progress beyond this stage now stands at 14 years, initially looked to up the tempo and get at the National League leaders, with Gray diverting Barton's cross wide with a first-time finish early on.
That pressure could not be sustained, however, and Lincoln were beginning to see more and more of the ball in a half that truly sparked to life midway through.
Barton was at the centre of the controversy, seemingly stamping on Matt Rhead's foot and then claiming that he had been elbowed, but replays showed that no contact was made.
The Englishman was also cautioned for flattening Terry Hawkridge later in the half, though chances were still proving to be hard to come by at Turf Moor.
Gray, who has seven goals in 22 outings this term, looked to be Burnley's most likely match-winning hero in the final quarter of the game, only to again squander a chance when spotted by Tendayi Darikwa in the box, while Ashley Westwood was denied by Paul Farman from the rebound.
Arfield was the next to send a shot wide, moments before Gray got in behind but saw his latest effort blocked aside, but there was time for one almighty twist at the death as Raggett found the key breakthrough goal.
A late corner was met by the centre-back - a scorer against Altrincham in the qualifying rounds - who was able to get enough power on the ball to get the better of Tom Heaton and send the travelling Imps fans into absolute dreamland.