The National League leaders made history as they became the first non-league club to reach the quarter finals of the competition since Queens Park Rangers in 1914.
Sean Raggett headed past Tom Heaton in the 89th-minute to secure the the historic win just in the nick of time at Turf Moor.
Burnley, who boast the third best home record in the Premier League, were unable to break deadlock with Andre Gray and Scott Arfield coming close to netting before Raggett’s winner.
The win comes after initially having to beat Guieley in a replay in October.
Lincoln have impressed on their route so far, knocking out Ipswich, Brighton and now Burnley in the process.
Football fans took to Twitter to express their jubilation as their team saw off a side 81 places above them in the football pyramid.
“Unbelievable Lincoln so proud! What a club! What managers and fans!” said one, while another added: “WOW #Lincoln. . . The whole country now wants you to win the FA cup! Even the Burnley fans.”
A Burnley fan wrote: “Can’t knock Lincoln today, deserved it all the way after the way the played and fair play to there fans as well, best at turf this season.”
Others posted videos of the jubilant scenes at the stadium as Lincoln’s 3,210 travelling fans celebrated the historic win.
