Albert Jarrett broke non-league Nuneaton's hearts with an 89th-minute free-kick to secure a 1-0 win for Lincoln.
The League Two strugglers looked to be heading for a replay against their Blue Square North opponents after playing the whole of the second half with 10 men following Clark Keltie's dismissal.
Nuneaton created a string of first-half chances with Eddie Nisevic shooting inches wide after just 30 seconds.
Midfielder Justin Marsden also went close after cutting through the heart of the Lincoln defence and full-back James Armson poked the ball wide after a 30-yard run.
The visitors continued to push forward after Keltie saw red for a foul on James Armson, with Gareth Dean shooting into the side-netting during first-half injury time.
Despite being a man down, Lincoln looked the stronger in the closing 20 minutes with Jarrett seeing a 25-yard shot come back off the post before he finally beat Nuneaton keeper Danny Alcock with a left-foot free-kick.
Lincoln keeper Joe Anyon still had to make a fine save to deny Nuneaton substitute Donovan Simmonds.