Jones' second half curler sent Anfield into ecstasy as the club's youth academy graduates handed Everton an embarrassing derby defeat.
The 18-year-old Liverpool-born midfielder was making only his fifth appearance and his memorable matchwinner was his first senior goal.
Jones is the youngest goalscorer for Liverpool in a Merseyside derby since Robbie Fowler in 1994.
“They played brave football. Unbelievable individual performances from the kids and the adults as well,” Liverpool boss Jurgen Klopp said.
“I'm so happy they all showed up tonight. Sensational game and a sensational goal from a Scouser. Who could ask for more?”
Without a win over their rivals in any competition since 2010, and thrashed 5-2 at Anfield in the Premier League earlier this season, Everton will feel the pain from this defeat against their under-strength neighbours for some time.
Hit by injuries that left him with 12 fit senior outfield players, Klopp had made nine changes, selecting Nat Phillips, 22, and teenage trio Neco Williams, Harvey Elliott and Jones.
In contrast, Everton manager Carlo Ancelotti made just two changes, fielding a strong team as the visitors chased a first win at Anfield since 1999.
Liverpool's injury problems mounted within 10 minutes when captain James Milner limped off to be replaced by another teenager Yasser Larouci.
Ancelotti's side threatened early on when Dominic Calvert-Lewin's strike was pushed away from Adrian, who then repelled Mason Holgate's header and Richarlison's shot.
Liverpool's youngsters had been thrashed 0-5 by Aston Villa in the League Cup while the senior stars were away at the Club World Cup in December.
But they acquitted themselves much better this time and Japan striker Takumi Minamino, making his debut after joining from Salzburg, glanced a header narrowly off-target before Divock Origi's low drive was brilliantly saved by Jordan Pickford.
Jones' moment of magic came in the 71st minute when he played a chipped pass to Origi, took the return ball and curled a stunning effort into the top corner.
Elsewhere yesterday, Lucas Moura kept Tottenham's FA Cup hopes alive with the equaliser in their 1-1 draw at second tier Middlesbrough, while Callum Hudson-Odoi got back in the groove as Chelsea cruised to a 2-0 win over Nottingham Forest.
Ashley Fletcher put Middlesbrough ahead in the second half at the Riverside Stadium.
But Brazilian winger Lucas headed the equaliser to send the tie to a replay in north London later in January.
“When we were losing 1-0, at that moment I had the feeling of 'we are in trouble',” Tottenham boss Jose Mourinho said. “But then everyone reacted. I reacted. The boys reacted.”
At Stamford Bridge, there were signs that Hudson-Odoi might be emerging from his sophomore slump.
He opened the scoring against the second tier side in the sixth minute, just his second goal of the season and his first since September.
Blues boss Frank Lampard had made nine changes, but Chelsea doubled their lead in the 33rd minute when Hudson-Odoi's shot was parried straight to Ross Barkley and the midfielder poked home at the back post.
Crystal Palace became the third Premier League club to be knocked out this weekend as second tier Derby won 1-0 at Selhurst Park.
Jayden Bogle fizzed a cross to the far post and Chris Martin tapped into the net in the 32nd minute.
In a VAR first in England, referee Michael Oliver used the pitch-side monitor to change a decision, sending off Luka Milivojevic after initially booking the Palace midfielder for kicking out at Tom Huddlestone in the 63rd minute.
Sheffield United avoided a non-league upset with a 2-1 win against AFC Fylde at Bramall Lane.
West Ham saw off third tier Gillingham as second half goals from Pablo Zabaleta and Pablo Fornals clinched a 2-0 win at Priestfield.
Shot shy United splutter
On Saturday (Jan 4) Manchester United were forced to settle for a dour 0-0 draw at Wolves while holders Manchester City started their bid to retain the trophy with a 4-1 victory against Port Vale.
Ole Gunnar Solskjaer's troubled side didn't manage a single shot on target in a domestic game for the first time in five years.
Easily beaten by Arsenal in the Premier League on Wednesday (Jan 1), United once again struggled to find any momentum and now face an unwanted replay at Old Trafford later in January.
Defending the latest in a growing list of underwhelming displays, Solskjaer said: “I'm happier to be in the replay than going out. It's a fair result that we are both still in the cup.
“It's a tough place to come. They had chances. Sergio Romero made some great saves. He's putting pressure on David de Gea and that's what we want. We want the best players in the world here.”
Solskjaer made seven changes at Molineux with the League Cup semi-final first leg against Manchester City looming on Tuesday (Jan 7).
Knocked out by Wolves in the FA Cup quarter-finals last season, disjointed United nearly fell behind when Matt Doherty's volley was superbly turned away by Romero.
Brandon Williams had a strong penalty appeal rejected by VAR after the United left-back tumbled following a challenge with Leander Dendoncker.
United substitute Marcus Rashford hit the bar with his first touch, but Wolves nearly won it in the closing moments as Raul Jimenez smashed against a post, leaving Solskjaer's men with just two wins from their last five matches.
At the Etihad Stadium, City boss Pep Guardiola made seven changes, but Oleksandr Zinchenko got the scoring started against fourth-tier Vale after 20 minutes.
Port Vale striker Tom Pope had memorably mocked John Stones last year, claiming he would love to play against him after being underwhelmed by the City defender's performance during an England match.
And with Stones returning for City after a month out with a hamstring injury, the 34-year-old Pope backed up his social media taunts by equalising with a glancing header in the 35th minute.
Sergio Aguero restored order when he fired City back in front with his 15th goal of the season three minutes before half-time.
Taylor Harwood-Bellis sealed the win from Stones' flick in the 58th minute and Phil Foden got the fourth in the 76th minute.
“We are in the next round after an incredible amount of games. We played seriously. First half we struggled a bit but second half was better,” Guardiola said.
Villa, Brighton crash out
Aston Villa and Brighton suffered embarrassing exits as the Premier League clubs paid the price for making a raft of team changes.
Villa are languishing just above the relegation zone after a difficult first season back in the top-flight and boss Dean Smith prioritised that survival battle with nine changes in the 2-1 loss at second tier Fulham.
Anthony Knockaert gave Fulham the lead with a 54th minute curler as the former Brighton forward grabbed his first goal since September.
Anwar El Ghazi equalised for Villa in the 63rd minute, but the understudies couldn't hold on as Harry Arter won it with a 30-yard thunderbolt with 16 minutes left.
“The game was won by two wonder-goals,” Smith said. “I don't like losing in any competition, but everyone could see this wasn't top of our priority list, because we have had injuries to three key players.”
Brighton endured the same fate as Villa as Albion boss Graham Potter made seven changes from their New Year's Day draw against Chelsea and second tier Sheffield Wednesday took advantage with a 1-0 win at the Amex Stadium.
Adam Reach blasted Wednesday into the fourth round with a deflected long-range strike in the 65th minute.
Watford blew a three-goal lead in a dramatic 3-3 draw against third-tier strugglers Tranmere, while Newcastle were held to a 1-1 draw by League One minnows Rochdale.
Arsenal host Championship highflyers Leeds United this evening.