Hesketh was one of Claude Puel's eight changes as four academy players started the game.
Charlie Austin's fourth goal in three games got Saints off to a good start but it was Hesketh who stole the headlines with a fantastic performance.
The 20-year-old played a number of fantastic passes as well as scoring Saints' second goal of the night and his first senior goal.
The attacking midfielder said it was a "special" night for him after helping Southampton progress into the next round.
He said: "It’s a night I’ve dreamed of since I was younger. I could only go out there and try and take the opportunity and get a goal, it’s really special for me."
Hesketh celebrated his goal with academy graduates Harrison Reed, Matt Targett and James Ward-Prowse, a moment that the 20-year-old labelled as "surreal".
"To celebrate with people I’ve grown up with is really special. There’s not many other clubs that that happens at. It’s very surreal."
Hesketh made his senior debut in December 2014 in a game against Burnley in the Premier League. However, he was taken off injured in the first half and didn't make another start for the first team.
Hesketh said that his disappointing debut against Burnley match made him even more determined to get back among the first team.
"Getting a taste 20 months ago, it’s made me really determined to get back in and around it on a regular basis. I want to cement myself in the squad and team. I have to take every opportunity I can.
"You can show what you can do at Under 23 level but you've got to show once you get an opportunity that you can take it. There’s not loads of opportunities flying about so you have to grasp it when it comes."
Saints will host Sunderland in the next round of the EFL Cup after their 2-0 win over Crystal Palace.