The defender sent his effort high over the Kop after Liverpool goalkeeper Giorgi Mamardashvili had denied Chelsea debutant Emmanuel Emegha. Valentin Barco also struck the post as Liverpool eventually prevailed in a dramatic shoot-out.
The victory extended Liverpool's winning run to five matches in all competitions, while Chelsea suffered a second consecutive defeat following their Premier League loss to Tottenham.
The teams had been unable to find a breakthrough from open play during normal time, with both goals coming from the penalty spot in a lively finish to the first half.
Morgan Rogers gave Chelsea the lead in the 40th minute after Ryan Gravenberch fouled Moises Caicedo. Liverpool responded three minutes later when Fofana inexplicably handled Kostas Tsimikas's corner, allowing Cody Gakpo to equalise from 12 yards.
Chelsea defender Maxence Lacroix was forced off with an apparent hamstring injury shortly before half-time, with Levi Colwill replacing him.
Both teams created opportunities to win the tie after the break. Chelsea goalkeeper Mike Penders produced important saves to deny Ronald Araujo and substitute Bradley Barcola, while Mamardashvili kept out efforts from Rogers and Marco Palestra. Florian Wirtz also missed a chance from close range for Liverpool.
Liverpool manager Andoni Iraola was pleased with the performances of his younger players, having included Trey Nyoni, Rio Ngumoha, James McConnell and Lewis Koumas in his starting line-up.
"I think they did a good job," Iraola said. "They deserved the right to play because they were also very good against Spurs in the last round. We told them they are first-team players and we are confident if they have to play."
He added: "They showed us they had a very good level and they helped us a lot."
Iraola described the match as closely contested, with few clear chances for either side, but welcomed Liverpool's progression to the last eight.
Chelsea manager Xabi Alonso was disappointed to exit the competition but believed his team had performed well enough to compete at Anfield.
"We saw a proper cup game, intense and really close. Then penalties are open for everything," Alonso said.
"It was a solid performance. I saw a team competing and showing a good level. This is the least important competition for us and now the focus is on the Premier League."
Liverpool will now prepare for a quarter-final against Bournemouth, Iraola's former club, with the ties scheduled for the week commencing December 14.
The other quarter-final ties will see Fleetwood face either Everton or Newcastle, Manchester City take on Bradford, and Crystal Palace meet Sunderland.
Michael Carrick has insisted he remains focused on improving Manchester United and does not need assurances about his future, despite mounting pressure following a dismal run of results.
Liverpool host Arsenal at Anfield on Sunday in a Premier League clash that could have a significant impact on the early-season title race.
Manchester United have recorded an estimated 334 million pounds in transfer-related losses over the past five years, according to a study by the CIES Football Observatory.
Manchester City manager Enzo Maresca has warned his team to prepare for another difficult test against Brighton when the sides meet in the Premier League on Saturday.
