The Premier League champions were on the board through Xherdan Shaqiri, Takumi Minamino doubling the lead on 18 minutes, Curtis Jones scoring a pair in a four-minute span and Minamino adding a fifth on the hour mark.
Lincoln pulled one back through Tayo Edun before Marko Grujic made it 6-1, Lewis Montsma scoring a minute later for the Imps and Divock Origi closing the scoring a minute from time as Liverpool set up a fourth round tie with Arsenal.
Story of the game
The first competitive meeting between the two sides in 59 years began the way it was expected to as Liverpool opened the scoring after just nine minutes when Shaqiri hit a sweet free-kick into the top left corner of the goal to give the Reds a 1-0 lead.
The visitors almost had a second soon after when Jones clipped his shot inches wide from a lovely ball sent in by Kostas Tsimikas. It was two on 18 minutes after Montsma lost possession and Minamino unleashed a first-time effort past Alex Palmer.
Lincoln did go close when James Jones shot across the face of goal, but Adrian collected with ease. Just before the half-hour mark, TJ Eyoma headed just over, causing a worry or two for the Premier League champions.
Two minutes following that chance, Rhys Williams sent a ball cross-field where Origi knocked it down and Jones showed some nifty footwork to get into position to beat Palmer into the top corner to make it 3-0.
Jones added Liverpool's fourth with a swift and brilliant move, spinning into the edge of space and stroked the ball into the far corner to put the game out of reach.
Lincoln kept battling despite the deficit on the scoreboard and Max Melbourne met Edun's cross on the right, but he could only head into the side netting.
It was 5-0 after halftime when Palmer saved from Jones, but Minamino was alert to a loose ball and bundled it over the line for second on the night.
The Imps had a trio of chances to get on the scoreboard with Harry Anderson, Jones and Edun all in the same sequence, but Adrian and his back line were there to see off the danger.
The hosts kept pressing and Jorge Grant stung Adrian's palms with a shot, Anthony Scully unable to convert from close range. Edun did get the Imps on the board, finishing from 12 yards to make it 5-1.
Grujic made it 6-1, beating Palmer from the 18 yard line and Montsma answered for Lincoln, a header ringing the crossbar and past Adrian to make the score 6-2.
Liverpool closed the scoring when Minamino played in Origi, who pummeled one into the far corner to send the Reds to the fourth round.
Man of the match: Curtis Jones
In a match that showcased Liverpool's depth, it was the performance of 19-year old Jones that stood out. His first goal was a combination of awareness to drag the ball back, took one touch and scored. His second was of similar skill, curling one past Palmer. The Reds have an embarrassment of riches and Jones, improving every match, is at the heart of that.
Liverpool have plenty of options going forward
While none of these players will be starters for Jurgen Klopp, as mentioned above, the depth shown off is going to make the manager very happy. Shaqiri hit a beautiful goal, Williams was leading the line in the first half and Tsimikas was plenty involved. Should Klopp need answers from his second unit, he now can call on these guys after performances like this.
Lincoln played an excellent second half
All credit to Michael Appleton's side for not giving up after falling behind 4-0 at halftime. The Imps had seven shots on target in the second 45 minutes and were duly rewarded with a couple of goals. It's not often Lincoln will play in front of the Sky Sports camera, but, despite the scoreline and obvious gulf in class, they can be proud of their effort.