The Owls were indebted to fine goalkeeping from Joe Wildsmith, who produced a number of strong saves to keep the League Two side at Bay.
Harrogate had 10 shots on target to Wednesday’s seven but it was the hosts who made it count with goals arriving from Saido Berahino, George Byers, Sylla Sow and Korede Adedoyin.
The Owls will play a runner-up from one of the seven other northern groups in the next round of the competition, with the draw set to take place on Saturday afternoon.
“We got what we wanted and that’s to top the group. That was the objective. It’s a really pleasing night for us, it’s another win and it keeps that momentum going,” said Moore, who made 10 changes from the side that drew 0-0 with Plymouth Argyle in the FA Cup on Sunday.
“We showed that ruthless streak I have been wanting to see and credit to the players for that. Harrogate came for us tonight but we got the first goal and after that we’ve seen some really clinical finishing.
“We couldn’t put out all the seniors tonight because of the turnaround of games but we have some very good players in the squad, some very good young players, and there were some very good performances out there tonight.”
Wildsmith’s performance on Tuesday came at an ideal time, with the 25-year-old in line to feature for the Owls when they host Gillingham in League One on Saturday.
Bailey Peacock-Farrell is on international duty with Northern Ireland, with the Green and White Army facing Lithuania on Friday night and Euro 2020 champions Italy on Monday in their final two World Cup qualifiers.
Those fixtures mean that Wildsmith is also likely to be part of the Wednesday side that faces Plymouth in an FA Cup first round replay on Tuesday.
“Joe was in tip-top form. He made a lot of those saves look easy and that’s because his positioning was spot-on,” added Moore.
“Bailey is away on international duty with Northern Ireland, Joe has come in, he is playing, and we have no problem with that.
“We are fortunate enough to have two fantastic goalkeepers who are neck and neck and pushing each other all the way.”