But Quincy Promes halved the deficit and Van Dijk thrashed home a stoppage time volley past Manuel Neuer to level.
That gave the Netherlands seven points in Group 1 of League A. It was the same tally as the world champions France but the Dutch advance to next summer's semi-finals on goal difference.
"We should all be proud of ourselves," said van Dijk. "I know I am. We worked so hard every game and to get rewards for that is fantastic."
While the Dutch join England, Switzerland and Portugal in the last four of League A, Germany will live to rue a plethora of missed chances after surging into a 2-0 within 20 minutes thanks to goals from Timo Werner and Leroy Sane.
Joachim Loew's men will be relegated from the group to continue the misery shrouding German football since the World Cup in Russia where they were eliminated after the group stages for the first time in 80 years.
The Dutch, by contrast, seem to be emerging from their slough of despond. They failed to qualify for Euro 2016 and this year's World Cup but have been reenergised since Ronald Koeman took over from the catastrophic two year reign of Danny Blind.
"If we can win the Nations League, fantastic," added van Dijk. "But we all want to make sure we're at the European Championships in 2020 and do very well there to make our country proud."
Koeman, a former Netherlands international, praised his team for their marksmanship in the closing stages. "The Germans were sharper and faster in the first half," he conceded. "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.
"Of course, we did not want to go down to League 'B', but winning the group was a dream."
Loew bemoaned his side's lack of precision. "That's the price you pay for having a young side," he said. "We have to learn to close a game down. The Dutch 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."