Harrison, linked with a £20million move to Premier League rivals Leicester, gave Leeds an interval lead with a 25-yard rocket before two second-half goals in as many minutes from Junior Firpo and Luis Sinisterra killed the tie.
Accrington teenager Leslie Adekoya scored within a minute of stepping off the bench to deservedly pull one back for the Sky Bet League One side.
When asked about Harrison's future at Elland Road, Marsch, who embraced his player after the final whistle said: "Listen, all I'll say is, I really like Jack first as a person. He's one of the most incredible people I've ever coached.
"We like him here. We want to keep him. He's performing really well. I think getting him fit and healthy has been a big key. He's got flexibility in the way he plays."
Leicester have been reported to have made a £20m bid for Harrison, whose Leeds contract expires at the end of next season.
Marsch said: "You know, when things start circulating, it's never so easy to just stay focused on what your job is, but I've tried to have open dialogue with him and be very honest with him and we have that kind of relationship.
"And then, you know, he does like it here. You know, that's all I can say. It's not like he's itching to leave."
Leeds sealed their place in the fifth round for the first time in seven years and for only the fourth time in two decades and Marsch is hoping for a home tie.
"Yeah, that's what I think would be really great, to play at Elland Road in such an important match and get our fans involved and get everybody excited in the city.
"We're happy that the team is coming together. We're happy that we've been able to push in this tournament and obviously home draw would be great."
Accrington boss John Coleman felt his side put up a good fight, but tired in the closing stages after playing 120 minutes on Tuesday night in their third-round replay against Boreham Wood.
Coleman said: "They're a top team Leeds aren't they? You wouldn't have known there were two divisions between us in the first half.
"But I just think their quality showed in the last 35 minutes and that's when the game was won and lost really.
"If we can play like we did for an hour in the rest of our league games we should have a good chance of staying up."
Coleman added: "Just because someone drives a better car and lives in a bigger house than you, it doesn't mean they're going to be better on a Saturday.
"But Leeds are a very good side and when they get it right they can blow teams away."