Lincoln, along with fellow National League team Sutton United, had ensured this was the first time since 1888 that two non-league sides had reached the fifth round.
The Imps are top of their league but 81 places behind Burnley in the football pyramid, and it was their determination to make another piece of history that saw them become the first non-league team since QPR in 1914 to reach the quarter-finals.
Danny Cowley's side have won 15 of their past 18 matches in all competitions and have now beaten Oldham, Ipswich and Brighton in this season's cup run.
This latest victory - assured and composed - means they are the first non-league side since Telford in 1985, and only the third ever, to knock out four league clubs in a single season.
As they celebrate their historic win, there will also be the realisation that they are just one win from a semi-final at Wembley.
Barton and Rhead do battle
Burnley have now lost five times in the fifth round of the FA Cup, having failed to reach the sixth round every season since 2002-03.
A strong home record of three losses in their past 29 matches at Turf Moor had put Burnley in a strong position. But Lincoln's tactic of blocking the hosts in numbers led to an engrossing battle between Joey Barton and Lincoln striker Matt Rhead, with the two players outmuscling one another throughout the 90 minutes.
Rhead, a domineering figure, indicated that Barton had stamped on him when the two met in a tussle inside the Burnley box, with Barton left on the floor holding his head.
The resulting tension led to an unsavoury scuffle involving nearly every player inside the Lincoln box. Barton was shown a yellow after catching Terry Hawkridge in the face with a shove, with the referee asking both benches to calm down.
The visitors capitalised on Burnley's frustration and the hosts were forced to shank several efforts away as Nathan Arnold kept crossing into the box.
Ultimately it was Cowley's pre-match mantra of playing positively that led to Lincoln keeping their cool in a frantic final few minutes.
There was a pause as players turned to the linesman to check that Raggett's sharp header had beaten the keeper and crossed the line. And when it became clear it was a goal, Lincoln players piled on their goalscorer as they celebrated a historic victory.
Man of the match - Matt Rhead (Lincoln City)
'It's a football miracle' - what the managers said
Lincoln boss Danny Cowley speaking to BT Sport: "It's a football miracle for a non-league team to be in the last eight. The boys were excellent, playing against a Premier League team.
"We are always realists, I will never ask the players to do something they're not capable of. But we thought if we could get the game plan right we could compete.
"Maybe young top-level centre halves are not used to playing against that up top. It was a bit of a throwback."
Burnley boss Sean Dyche speaking to BBC Sport: "I'm only gutted in the sense that could happen, I was on the other side of it [at Chesterfield]. You don't want to be part of it for the wrong reasons.
"They used everything they needed to use. They played our pitch well, played resolutely with their back four not going anywhere and they got the ball forward.
"My team were nowhere near the level they can show. You have to work, be diligent and believe you will get another chance. I think they only had one chance, credit to them."
One shot on target, one goal - the stats
Lincoln City are the second non-league side to knock out Premier League opposition, after Luton Town (v Norwich City in January 2013).
The last non-league side to be in the last eight of the competition were QPR in 1913-14 [round four].
Two of the past four occasions of a non-league side winning away against a top-flight team have come against Burnley [also Wimbledon in January 1975].
The Imps are only the third non-league team to keep a clean sheet on an away Premier League ground, following Exeter City at Old Trafford against Manchester United in January 2005 and Luton Town at Carrow Road against Norwich City in January 2013.
The past two occasions a non-league team have beaten a Premier League team, Andre Gray was involved, for Luton in their win versus Norwich in January 2013 and for Burnley in their defeat against Lincoln in this match.
It was Sean Raggett's second FA Cup goal this season, having also scored against Altrincham in the first round.
Raggett's 89th-minute winning goal for Lincoln was their only shot on target in the game.
Lincoln, who find out who they face in the quarter-finals on Sunday, are away at North Ferriby United in the National League on 21 February. Burnley return to Premier League action on 25 February away at Hull.