Sean Raggett's goal in the 90th minute settled a scrappy match at Turf Moor, making Lincoln the first minor league club to reach the last eight of English soccer's most prestigious knockout competition since 1914.
The visitors, who play in the fifth-tier National League, matched a side 81 places higher up the English ladder and should have taken an early lead when Jack Muldoon fired over.
Burnley, with only three defeats in their previous 29 home matches in all competitions, struggled to make their higher status show until the latter stages when they finally began to pile pressure on a tiring Lincoln side.
But just when it seemed the Premier League side would score, Lincoln were awarded a corner and after Burnley failed to clear Raggett rose above a crowd of players to head the ball over the line.
They survived five minutes of stoppage time, including Burnley's keeper Tom Heaton going forward to try to salvage a draw, to spark wild celebrations among the 3,500 travelling fans.
Manchester City travel to Championship Huddersfield Town and Premier League leaders Chelsea go to second tier Wolverhampton Wanderers in fifth-round ties later on Saturday.
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