Elsewhere, Harry Kane scored twice as Bayern Munich downed Club World Cup winners Chelsea 3-1 as Pafos and Bodo/Glimt marked their Champions League debuts with hard-earned draws.
Liverpool, with club-record signing Alexander Isak on debut, raced into a two-goal lead inside the first six minutes as Atletico struggled to get out of their own half on Merseyside.
A Mohamed Salah free-kick flew in off Andy Robertson for a fortuitous fourth-minute opener, before the Egyptian forward collected Ryan Gravenberch’s pass and slotted into the bottom corner.
But Atletico eased themselves into the game and halved the deficit in first-half injury time.
Marcos Llorente, who scored twice in Atletico’s famous last-16 win at Anfield five years ago, stabbed the ball beyond a diving Alisson Becker from the edge of the box.
Liverpool missed several chances to restore their advantage, including a glaring miss from Salah when he struck the post, and Llorente made them pay with a deflected late volley to draw Diego Simeone’s men level.
But Arne Slot’s English champions have won all four of their opening Premier League games of the season with winners inside the last 10 minutes.
This time it was skipper Van Dijk who produced the big moment with an excellent header from Dominik Szoboszlai’s corner in the 92nd minute.
“We found a way again. Glad we got the win,” Van Dijk said. “Today I was the lucky one with the winner and we move on.”
Atletico coach Simeone was shown a red card after a confrontation with Liverpool fans following Van Dijk’s goal.
Meanwhile, PSG got their first Champions League campaign as reigning European champions off to a flying start with a 4-0 win over Atalanta in Paris.
Captain Marquinhos tapped into an empty net in the third minute, before Khvicha Kvaratskhelia, who was an injury doubt in the build-up, crashed home a fine second for Luis Enrique’s men six minutes before half-time.
It could have been even better for the hosts at the Parc des Princes if not for Bradley Barcola’s tame 44th-minute penalty which was comfortably held by Atalanta goalkeeper Marco Carnesecchi.
Atalanta never looked likely to stage a comeback, though, and Nuno Mendes added a third not long after the restart.
Substitute Goncalo Ramos rounded off an excellent night for PSG with a dinked finish in stoppage time.
“It was perfect, but obviously the most important thing was the team winning. We played a great game,” Kvaratskhelia said. (AFP)