The draw was evenly balanced until Jones – a product of the academy that made its first team debut last year – closed the competition with an excellent curling shot from the edge of the area.
But thanks to Jones' goal and three saves by goalkeeper Adrian in the first half, Klopp's risk paid off.
"The ball came to me and I had only one thing in mind – to shoot. Fortunately, it led to the winner and a great goal," said Jones.
"It's huge for me and the rest of the boys. Sometimes it's frustrating to think that you might get a chance, but then you have to sit on the bench and watch.
"We have proven that we learn every day and try to secure a place in the team."
It was a second consecutive defeat under new coach Carlo Ancelotti for Everton, who had started the game brilliantly in Anfield.
Dominic Calvert-Lewin, Mason Holgate and Richarlison provided an excellent parade for Adrian in the first half when the guests checked the Liverpool goal with a number of test crosses.
Red captain James Milner was injured after six minutes when he refused Calvert-Lewin in the box. He was replaced by 19-year-old Algerian Yasser Larouchi. So four teenagers – including 16-year-old Harvey Elliott – stood on the pitch for Liverpool.
But despite all the chances of Everton, the youthful home team showed serenity in the first phase and had a lot of possession and territory.
After the break, Liverpool came out stronger and played at a faster pace. Meanwhile, Everton was dull.
Ancelotti introduced former Juventus striker Moise Kean to overcome the impasse. However, it was Liverpool and Jones’s day that the teenager scored his first goal in 71 minutes in great style.
The result means that Liverpool remain unbeaten in national games this season and have been unbeaten in Anfield since September 2018.
In the previous FA Cup games on Sunday, Tottenham drew 1-1 with the Middlesborough championship team.
Ashley Fletcher had given the home team, currently 16th in the second division of English football, a second-half lead before Lucas Moura's header caused the tie to be played again.
The other two Premier League teams that played on Sunday won when Chelsea defeated Nottingham Forest 2-0 and Sheffield United 2-1.