Van Dijk is set to miss the rest of the season with a knee injury but Jurgen Klopp’s side kept a clean sheet and took all three points after Nicolas Tagliafico turned a wayward Sadio Mane shot into his own net on 35 minutes.
“It was not sunshine football but we wanted three points and we got it,” said Klopp.
The 2019 European champions will hope to build on this result when they host Midtjylland next week.
The Reds, who also face Atalanta and Midtjylland in Group D, were returning to Europe as English champions for the first time since 1984-85.
The Danes were outclassed at home by Atalanta on Wednesday, losing 4-0 with Duvan Zapata, Alejandro “Papu” Gomez and Luis Muriel scoring in the first half and debutant Aleksei Miranchuk adding a late fourth.
Pep Guardiola’s City made it three wins for English clubs this midweek as they came from behind at home to beat Porto 3-1.
Luis Diaz gave Porto the lead with a superb individual goal but Sergio Aguero’s penalty soon brought City level and Ilkay Gundogan’s freekick put them ahead on 65 minutes.
Substitute Ferran Torres then scored a superb goal to make sure of the win.
Also in Group C, Egyptian substitute Ahmed Hassan headed in a stoppage-time winner as Olympiakos downed Marseille 1-0.