Jurgen Klopp's side were eliminated from Europe's secondary competition despite prevailing 1-0 in Bergamo, having fallen to a chastening 3-0 loss at Anfield in last week's first leg.
The Merseyside giants exploded out of the blocks and were rewarded for their quick start with just seven minutes on the clock, as Mohamed Salah crashed home a penalty after Matteo Ruggeri's handball from Trent Alexander-Arnold's cross.
The Egyptian squandered a brilliant chance to cut the aggregate deficit further later in the first half, though, as his lob over Atalanta goalkeeper Juan Musso - caught in no man's land - bounced the wrong side of the post.
However, Liverpool were a shadow of their first-half selves in the second 45, where they fired just two shots on the Dea goal for an Expected Goals (xG) of 0.11 to ultimately exit with a whimper.
Van Dijk praises "great fight" in Liverpool exit
Despite the lackadaisical nature of the visitors' second-half performance, Van Dijk - speaking to TNT Sports - believed that the Reds showed "great fight" to prevail on the night, even if their triumph was inconsequential.
"Firstly credit to Atalanta. Last week we were very poor and they had a good game. They deserved to go through because we made it far too difficult for ourselves," the Liverpool captain said.
"Tonight was a good game and we showed great fight. It was an improvement but the reality is we are out and we have to very quickly recover and travel to London on Saturday.
"We tried and we had opportunities. At times we wanted it too quickly as everyone was keen to get the second and third. But overall it isn't enough. We are disappointed to be out of the competition as we really wanted to win it."
What next for Liverpool after European heartbreak?
Despite their previous monumental fightbacks against AC Milan and Barcelona in the Champions League, Liverpool's penchant for turning ties around meant little at the Gewiss Stadium, where Klopp took charge of his final European game as Liverpool head coach.
However, the Reds - who have already been crowned 2023-24 EFL Cup winners - could still achieve springtime silverware, as they resume their Premier League campaign away to Fulham on Sunday afternoon.
Van Dijk issued a rallying cry to Liverpool's die-hard supporters ahead of the trip to Craven Cottage, claiming that the Reds now need their fanbase "more than ever", adding: "We won tonight and we kept a clean sheet, so there are so many more positives we can take. Yes, it is a bad feeling to be out so we have to get back up.
"We have been through difficult moments all together, so showing maturity and togetherness is the key to this. And professionalism to be ready for Sunday as that will also be a tough game. We all have a responsibility to be ready for it physically and mentally.
"We need everyone now for the last push in the league. Hopefully our fans will be there because we need them now more than ever because anything is still possible."
Liverpool occupy third place in the Premier League table with six games remaining, behind Arsenal on goal difference and two points worse off than leaders Manchester City.