It was a back-and-forth first half between the two sides on Tuesday, with Liverpool eventually striking twice in two minutes after the interval through Naby Keita and Mohamed Salah as they ended Salzburg's hopes of an upset.
Despite some nervy moments, Liverpool handled the game well in the second half and saw out the result, qualifying for the knockout stages of the Champions League with a mature performance.
Speaking after the game, Reds defender van Dijk said: "We had to be ready - and ready for a very tough night because they are a good side.
"They play a little bit similar to what we try to do, direct and try to get in behind the defence and stretch them. They did well.
"We had to be ready and we dealt with it and then obviously at the end of the first half we had the opportunities to score for 1-0, maybe 2-0 with Naby but unfortunately we didn't do that.
"In the second half we were very mature and played a great second half. The belief is always there but you have to show how much you really want it.
"I think especially the second half we can look back good on. The goals we scored were very good, the whole second half I think everyone looked comfortable.
"Obviously at times they broke on us and that's the quality they have as well and you have to give them credit for that, too, and we dealt with it pretty well."
The Dutchman also praised Salah for showing his "true quality" in scoring an extraordinary goal, although he admits the Egyptian "could have made it a little bit easier" for the Reds after missing opportunities earlier in the game.
"It was a big night for [Naby] and I'm very happy for him to be that important and hopefully he can keep being important like the rest of the boys," he said.
"He's a very good player and we all can see that and today he showed it with the goal as well.
"Sadio was outstanding, Bobby did so much work as well and it's a team performance, especially the second half.
"[Jordan] is a fantastic player, a fantastic captain and everyone is looking up to him and he is very important for us.
"Like him, we have everyone that we need, and today he was very important with the assist and also in the game as well."
Liverpool are next in action at lunch-time on Saturday as they welcome Watford to Anfield for another crucial Premier League encounter.