Mohamed Salah finished with aplomb in the 23rd minute after James Milner stole possession from Yves Bissouma but the Reds did not pull away as they might have hoped against a Brighton side riding high on last weekend's 3-2 win over Manchester United.
As in the 2-0 win at Crystal Palace on Monday, Jurgen Klopp's men showed a resilient side that pleased centre-back and record signing Van Dijk.
"It was a tough game, we knew before the game Brighton were full of confidence, we just had to be ready for the battle," he told BT Sport.
"We didn't play our best game but it's very important that we won and kept it tight in the end.
"We [defend] all together. It starts up front, you can see how important it is.
"Credit to everyone, we're not doing it alone at the back, everyone's doing their bit.
"It's always important, those crucial saves, so we're happy to keep a clean sheet and win."
Salah now has 29 goals in as many appearances at Anfield, leaving Milner to marvel at his prolific team-mate.
"Not a bad record is it. If Mo gets those chances he'll take them," he told BT Sport, before sending further praise in the direction of Van Dijk and the defence operating in front of Alisson – the Brazil goalkeeper already proving his worth after a big-money arrival from Roma.
"The front three are always going to cause problems for most teams, that’s always going to get the headlines," he added.
"But the boys at the back, the goalkeeper and as a team keeping these clean sheets is so important. That's what's getting us the points."