The point in Gelsenkirchen was enough for the Dutch to advance to the last four at the expense of world champions France after scoring twice in the final six minutes in a dramatic finish.
Liverpool defender Van Dijk volleyed home a cross in added time to complete the Dutch fight back after Quincy Promes had scored their first goal on 85 minutes.
Germany, who led 2-0 after only 20 minutes through goals by Timo Werner and Leroy Sane, were punished for not being clinical enough in the second half.
Van Dijk said it was "a fantastic feeling" to finish top of Group One ahead of European powerhouses France and the Germans.
"We should all be proud of ourselves. I know I am," he said after the Dutch join England, Switzerland and Portugal in the Nations League's last four.
"We worked so hard every game and to get rewards for that is fantastic."
After missing out on qualifying for the finals of both Euro 2016 and this year's World Cup, Van Dijk said the priority is to get the Dutch to the Euro 2020 finals.
"If we can win the Nations League, fantastic, but we all want to make sure we're at the European Championships and do very well there to make our country proud," he added.
Netherlands head coach Ronald Koeman praised his team for turning two of their three late chances into goals.
"It was very difficult for us, the Germans were sharper and faster in the first half," admitted Koeman.
"We fought to come back in the second half and pressed.
"They had their chances to finish it off, but the fight and belief we showed was enough for the point.
"We didn't play great football, it wasn't our best performance, but it was enough.
"No one expected Holland to win the group.
"Of course, we did not want to go down to League 'B', but winning the group was a dream."
Germans rue wasted chances
After Germany missed several second-half chances, head coach Joachim Loew rued the victory that got away from his inexperienced side.
"That's the price you pay for having a young side, we have to learn to close a game down," said Loew, who started only three 2014 World Cup winners in Mats Hummels, Toni Kroos and Manuel Neuer.
"They risked everything and were rewarded.
"This is a learning experience for us and we will have to analyse our performance in the last 10 minutes.
"We showed for 80 minutes that we are strong and they hardly had a chance to score, which says a lot for our team."
Thomas Mueller reached the milestone of 100 Germany appearances as a second-half replacement, but the future belongs to the likes of Manchester City's German winger Sane, who was outstanding.
"The result is annoying," Sane said.
"We went ahead and then let them off the hook.
"We should have scored the third goal, then you run the game down, but it didn't work out like that."
Fellow goal-scorer Werner, who ended his eight-game goal drought, agreed.
"It's a pity, we had everything under control until the 85th minute and could have gone 3-0 up," said Werner.
"At the end, we weren't rewarded and everything fell apart again."
Despite finishing the year with a 3-0 win over Russia last Thursday and a draw with the Dutch, 2018 has been a nightmare year for the Germans.
They have suffered a record six defeats in the last 12 months, exited the World Cup bottom of their group and are relegated to the 'B' division of the Nations League.
Yet Loew, who has a contract until the 2022 World Cup, has kept his job and looks safe until the Euro 2020 finals.
"It's been a very disappointing year after the World Cup didn't go the way we wanted it to.
"However, I have the feeling we are on a good path with these players and will be successful again with this young team."
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