The Reds won 2-0 thanks to a brace from Mohamed Salah.
Van Dijk said afterwards: "Very good win, clean sheet, hard work for both sides - it was going to be tough.
“Bournemouth in red-hot form, it was always going to be difficult.
"We created good moments, and I think it went a bit how we expected. There are times you could maybe be better - making the right decisions with the ball. But overall we cannot complain - 2-0 where a lot of teams have dropped points."
On Alisson, he said: “There were good saves, maybe even more.
“We don’t want our goalkeeper to work hard, but today was always going to be difficult as they have the quality to hurt you.
“We needed him, and he deserves a nice evening off tonight.”
On Salah, Van Dijk also stated: "Let's make it more, let's keep going. He has obviously been amazing for the football club over all those years.
"I see the hard work he puts in, but the numbers speak for itself.
“It's an outstanding performance, but he is a phenomenal player.”