Liverpool progressed to the quarter-finals of the competition with a 3-1 win at the Allianz Arena, thanks to a Sadio Mane brace and Van Dijk header.
After a goalless draw in the first leg at Anfield, many experts tipped the Reds to suffer an early European exit in Germany, but they completely outgunned Bayern on the night.
Van Dijk delivered yet another typically assured display at the back and his 69th-minute goal ultimately turned out to be the winner.
At a post-match press conference, Klopp hailed the impact Van Dijk has had on his squad before also insisting that Liverpool did not reach their best level against the Bundesliga giants.
The German boss said: "It was brilliant. He was involved in the goal of Bayern as well, so what a night! What can I say?
"I could write a book about his skills, his strength, how much I like him, what a fantastic person he is.
"He is so young, already so mature, really strong, all good. Virg knows he can play better than he did tonight; we can play better than we did tonight.
"Tonight we played - especially the second half - as good as necessary and as possible.
"It is a difficult place to come and not a lot of teams have on their cards an away win at Munich. We have that now and that's cool. Difficult but cool."
The Reds must now shift their focus back to the Premier League title race, with a crucial fixture against Fulham on the cards this Sunday.
Klopp's men are a point behind reigning champions Manchester City in the table at the moment, but they can steal a march on their rivals this weekend while Pep Guardiola's team negotiate an FA Cup quarter-final.
Liverpool can also start looking forward to a Champions League quarter-final, with the draw for the last eight set to take place on Friday morning.