The National League leaders dominated lengthy spells of the game but had to wait until the 91st minute to net the game's only goal through Nathan Arnold.
Theo Robinson proved a handful for the Tractor Boys in the first half and he almost fired the Imps in front in the 23rd minute, turning in the box and letting fly, only to be denied by a crucial block from Paul Digby.
The home side came even closer on the stroke of half time when Matt Rhead's knockdown fell kindly for Lincoln skipper Luke Waterfall to strike inside the box, with Dean Gerken forced to meet the effort with a reaction save.
Mick McCarthy's men were less submissive in the second period, and they almost broke the deadlock against the run of play on the hour as Josh Emmanuel's strike from outside the box was spotted late by Paul Farman in the Imps goal, and they quickly recovered to palm the ball away.
With 90 minutes on the clock and both sides fatigued, Ipswich won a free kick deep in the Lincoln half.
However, a poor delivery from Matt Lawrence gifted possession back to the opposition and substitute Adam Marriott broke at pace before threading the ball through to Arnold, who rounded Gerken and fired home to send Sincil Bank into a frenzy.
Lincoln will receive £67,500 in prize money for dumping the Tractor Boys out of the FA Cup and play host to Brighton & Hove Albion in the next round.