Jurgen Klopp made 10 changes from the side that beat Chelsea 2-0 in the Premier League on Sunday, with only Virgil van Dijk keeping his place, but they still had more than enough firepower to swat aside their League One opponents.
Xherdan Shaqiri got the visitors on their way inside the opening 10 minutes, whipping a superb free-kick into the top-right corner, before a fine strike from Minamino and a double from Jones made it 4-0 at the break.
Minamino added a fifth immediately after the restart before Tayo Edun pulled one back for the Imps on the hour mark.
Marko Grujic and Lewis Montsma then exchanged quick-fire goals, while Divock Origi added a seventh late on as Liverpool – who handed Diogo Jota a debut from the substitutes' bench in the second half – set up a fourth-round clash with Arsenal at Anfield next week.
Liverpool, who included debutants Rhys Williams and Konstantinos Tsimikas in their starting XI, wasted little time asserting their authority as Shaqiri registered his first goal since December 4 with a sublime free-kick from 22 yards in the ninth minute.
The Reds doubled their advantage after 18 minutes when a sloppy pass out from the back by Montsma found its way to Minamino, who sent a strike into Alex Palmer's top-left corner from 20 yards.
Jones added a third shortly after the half-hour mark, curling home from inside the penalty area after being picked out by Divock Origi's downwards header.
The 19-year-old claimed his second just four minutes later when his effort from outside the area clipped off Timothy Eyoma and flew past Palmer.
Any thoughts that Klopp's men would take it easy in the second period were extinguished after just 18 seconds when Minamino volleyed into an empty net after Harvey Elliott had been denied by Palmer.
The Imps started to pour forward in search of a consolation and it duly arrived on the hour mark when Edun coolly slotted past Adrian from inside the area.
Grujic plundered his first Liverpool goal from 20 yards in the 65th minute before Montsma took advantage of Adrian's woeful goalkeeping to head home Lincoln's second just a minute later.
Origi then wrapped up the scoring a minute from time when his strike from 15 yards slipped through the grasp of Palmer.
The 19-year-old was superb in Liverpool's 7-2 EFL Cup third-round win over Lincoln City on Thursday, scoring twice as the Premier League champions set up a fourth-round clash with Arsenal next week.
It was only Jones' 16th appearance for Liverpool's first team but Klopp suggested the midfielder can expect to feature more regularly if he continues to produce such impressive displays.
"I think each game is ideal for Curtis to play in because of the quality he has, the skills he has," Klopp told a media conference.
"First half, really good. Start of the second half, really good. And then there is still space for improvement, obviously.
"Both goals were difficult. I think the second goal, the first touch was absolutely exceptional. It was slightly deflected, but that's just exceptional.
"I think everybody knows my opinion about Curtis. He is an exceptional talent and we will have some fun with him, I'm pretty sure, in the future."
Takumi Minamino also impressed for Liverpool, adding to Xherdan Shaqiri's opener with a sublime effort from outside the area and then claiming his second shortly after the interval.
While Klopp was pleased with his potency in front of goal, he was also quick to praise the Japan international's relentless energy.
"I loved Taki's game tonight; it was absolutely outstanding and he was involved in so many situations," Klopp added.
"More and more, I think we all realise where his qualities are. Wherever he will play in the end, he will end up in the centre. If he plays on the left wing or the right wing, he will play in the half-spaces or the centre because that's where he has his real strength.
"The opponent had to kick off seven times [after Liverpool's goals] and the first guy who chased the opponent again when they received the pass was Taki.
"He set the rhythm for all these pressing and counter-pressing situations, so I really, really liked his game tonight."
Liverpool face Arsenal in the Premier League on Monday before hosting them again at Anfield later in the week, although no date has been set yet.
In the standout third-round tie of the day, Liverpool host their local rivals at Anfield and Jurgen Klopp has made the most of the opportunity to rest key players, as he had previously suggested.
But among those to profit is new signing Minamino, who joined at the start of the month following a reported £7.25million move from Austrian side Salzburg.
The Japan international had impressed in two Champions League matches for Salzburg against the Reds in the first half of the season, scoring in the 4-3 defeat to Klopp's men in October.
He was ineligible to feature in Liverpool's 2-0 win over Sheffield United on January 2, but following confirmation of his registration, the 24-year-old comes straight into a starting XI that does not include Roberto Firmino, Virgil van Dijk, Mohamed Salah or Sadio Mane.
Nathaniel Phillips, Neco Williams, Pedro Chirivella, Curtis Jones, Harvey Elliott and Divock Origi get rare starts for the Reds, while Everton have named a line-up almost at full-strength in Carlo Ancelotti's first Merseyside derby.
Richarlison, Theo Walcott and the in-form Dominic Calvert-Lewin lead the line, while Mason Holgate keeps his place at the back alongside Yerry Mina, with Michael Keane still benched.
