Mohamed Salah's first-half goal was enough for the Premier League leaders to edge Napoli 1-0 at Anfield on Tuesday.
But they needed some late heroics from Alisson, who made a good save to deny Arkadiusz Milik and help Liverpool hold onto second place in Group C.
Van Dijk, who is suspended for his side's next Champions League game after being booked for a dangerous tackle on Dries Mertens, praised the Brazil international goalkeeper.
"You don't think too much because it happens in a quick second," the centre-back told reporters.
"[It was a] fantastic save, very important for all of us and fantastic keeper."
THAT. SAVE. pic.twitter.com/mLIlI0euLA— Liverpool FC (@LFC) December 11, 2018
Despite the success, Van Dijk warned his team they still had a lot of work to do this season.
"We didn't achieve anything. We didn't achieve anything. We won the game, we're through to the next round and that's just the next step for us," he said.
"We want to achieve big things this season, like every year I think, and we're on a good way."
After a fourth successive win, Liverpool have another big game on Sunday, hosting Premier League rivals Manchester United.