Andoni Iraola's side booked their place in the quarter-finals on Wednesday after beating Chelsea 6-5 on penalties following a 1-1 draw during normal time.
It was Liverpool's fifth consecutive victory across all competitions and their third successive home success, with the captain believing the performances are helping restore the connection between the players and supporters.
The Reds had previously defeated Villarreal in the Champions League and Brighton in the Premier League at Anfield, providing further encouragement ahead of Sunday's meeting with Arsenal.
Van Dijk believes the supporters are responding to what they have been seeing from the team, with the atmosphere becoming an increasingly important part of their recent improvement.
"It's a night like this and obviously the other night [against Villarreal]," he said.
"I think the fans definitely like what they're seeing and, of course, you feel that energy and it's great.
"I've been feeling that energy for quite a while and when you then miss it a little bit then to get it back is obviously very good. Results help, massively. We understand it as well.
"We want to show that for the full 90 minutes, every game, home and away and obviously it's not the easiest thing to do but we're trying to and nights like this helps.
"We put a lot of work in behind the scenes. Hopefully the reward will come soon."
The Liverpool captain was particularly pleased with the team's performance against Chelsea, believing they restricted their opponents to relatively few clear opportunities despite the intensity of the contest.
The two sides exchanged penalties late in the first half, with Morgan Rogers giving Chelsea the lead before Cody Gakpo equalised three minutes later.
Neither team could find a winner during the second period, leaving the tie to be decided from the penalty spot.
Van Dijk also reflected on his own involvement, having returned to the starting line-up for a match against one of Liverpool's major domestic rivals.
"Was I surprised to start? No, no. I was really looking forward to starting, regardless," he said.
"I've been, like you said, only four times been having a call-up (prior to the semis) but to play at home against Chelsea and the team needed me so I'm just happy."
The defender was replaced by Jérémy Jacquet in the 72nd minute, having felt comfortable with his performance before coming off.
"I felt good after 70 minutes or so, just had a little bit of a rest. It was unfortunate to get the penalty against us. I don't think we gave much away throughout the whole game.
"A couple shots from distance, I think. Overall, we did well. We brought a lot of energy. Obviously, they have their patterns in terms of like, play and bounce and runners in between, and I think we did well.
"We could have done maybe a couple of times better. Overall, I'm happy with the evening and I'm happy that we went through."
Van Dijk also praised goalkeeper Giorgi Mamardashvili, whose contribution proved important during the shootout.
The Georgian denied Chelsea debutant Emmanuel Emegha from the spot before Joe Gomez converted for Liverpool. Wesley Fofana subsequently sent his penalty over the crossbar, confirming the home side's progression.
The captain was delighted for Gomez, who had taken responsibility at a decisive moment.
"Joe's [penalty] is amazing. I was so happy for him. But yeah, I think Giorgi was very important too, again.
"It's a great night for all of us, we got through, and now it's Bournemouth at home, so let's go."
He also singled out Mamardashvili for his performance throughout the contest.
"I think [Mamardashvili] was very good. I'm happy that he played a big part in us going through and it's great for him this evening, so happy for him. I'm happy for us as well. I think it was a good game and great to go to the quarter-finals."
The match also included an unusual incident involving Chelsea substitute Emegha, who shoved Van Dijk to the ground away from the ball during the second half.
The Liverpool defender avoided making a major issue of the confrontation when questioned afterwards.
"Ask him. I don't know.
"Maybe he's trying to get into the national team as well and trying this way. I don't know. You have to ask him. When he comes by, just ask him."
Liverpool will now turn their attention to Arsenal before preparing for their Carabao Cup quarter-final against Bournemouth.
For Van Dijk, the return of a positive atmosphere at Anfield is an encouraging development, although he knows the team must continue producing performances that justify the supporters' growing enthusiasm.