Tottenham Hotspur's Lucas Moura celebrates his goal against Middlesbrough
London - Jones is the youngest goalscorer for Liverpool in a Merseyside derby since Robbie Fowler
Jones' second half curler sent Anfield in 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 match winner was his first senior goal.
Jones is the youngest goalscorer for Liverpool in a Merseyside derby since Robbie Fowler.
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 while.
Hit by injuries that left him with 12 fit senior outfield players, Liverpool boss Jurgen 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 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 by Aston Villa 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 on Sunday, 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.
Tottenham haven't lifted a major trophy since 2008 when current Middlesbrough manager Jonathan Woodgate scored the League Cup winner against Chelsea.
Woodgate was on course to dent Tottenham's bid to end that wretched run when 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.
"We could be out like a few other Premier League teams. 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.
"I was particularly pleased for Callum, he has had a bad injury, is getting himself fit and getting sharper. His goal was an early settler for us," Lampard said.
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.