Fulham took a first-half lead in Wednesday's clash with a goal from Willian and then sat back and looked to hit Liverpool on the break, frustrating Juergen Klopp's side until the 68th minute when Jones and then Gakpo struck in quick succession.
"It's the mentality in the group, in the club, in the city, just keep going and it doesn't matter what happens and we showed that again today," Gakpo told Sky Sports. "If you have that mentality a lot of things can happen."
Fulham threatened to hand the record nine-time League Cup winners their first home defeat across all competitions this season after Willian scored in the 19th minute. Capitalising on Virgil van Dijk's poor clearance, the Brazilian slotted home from the edge of the six-yard box between keeper Caoimhin Kelleher's legs.
Premier League-leading Liverpool dominated possession and finally got the better of Fulham's well-organised back line when Jones netted, his shot from 25 metres out taking a big deflection off the back of Tosin Adarabioyo.
Substitute Gakpo, who has scored in all four of his League Cup games for Liverpool, fired the home team in front in the 71st, tapping in Darwin Nunez's cross at the near post to send fans home happy after they extended their remarkable home record this season to 14 victories and two draws.
Klopp's side had 21 shots to Fulham's six, and the addition of Gakpo and Nunez in the second half made all the difference.
The match marked the seventh time Liverpool have won after conceding the first goal in all competitions this season, the most of any side in England's top four tiers in 2023-24.
Liverpool were missing talisman Mohamed Salah, the Premier League's joint top scorer this season who is with Egypt at the Africa Cup of Nations, and Trent Alexander-Arnold, who injured his knee in Sunday's FA Cup third-round win over Arsenal.
Klopp praised the "exceptional" performance of 20-year-old Conor Bradley, who started in defence in place of Alexander-Arnold.
"He will probably fall asleep in the dressing room," Klopp said. "Wonderful, what a boy, what a character. Everyone loves him."
Liverpool will travel to Craven Cottage for the second leg on January 24.
"We knew it was going to be a tough one. You can't play for a final thinking it's not going to be really hard," said Fulham manager Marco Silva, whose Cottagers are making their League Cup semi-final debut.
"At home, with our fans behind us, it's going to be a great night for Fulham Football Club."
Chelsea suffered a surprise 1-0 loss at second-tier Middlesbrough in the first leg of the other semi-final on Tuesday night.
