Rangnick's men edged a dramatic closing Group D encounter 3-2 against the Dutch, who had to settle for going through in third place, while France were held to a 1-1 draw by already-eliminated Poland.
Austria jumped up from third before kick-off and will next face either Turkey, the Czech Republic or Georgia in the last 16 in Leipzig on July 2.
They have never reached the European Championship quarter-finals.
"We started in the tournament with a 1-0 loss with an own goal against France and if you knew the pressure we had against Poland (3-1 win) when we knew we had to win to have a realistic chance to qualify, then to finish top of the group is incredible," Rangnick told reporters.
"If you had bet on us to win here and France not to win you'd be a very rich man or woman...
"But that's the great thing about football and we were rewarded for a great and energetic performance from my lads."
Rangnick surprisingly made four changes from the team which beat Poland, knowing Austria would qualify unless they suffered a defeat by more than four goals.
He said Italy's last-gasp equaliser on Monday against Croatia allowed him to rest some players who would have picked up a one-match suspension with a yellow card.
"If the goal in the Italy game hadn't gone in in the last minute, we would've had a different line-up as we wouldn't have been basically qualified," added Rangnick.
"So we had an unexpected line-up but in the end all the thinking was worth it as we were able to go all in with this line-up and we didn't have to think about second bookings and so forth and it worked out."
Two of Austria's squad -- Nicolas Seiwald and Christoph Baumgartner -- play for RB Leipzig, while key midfielders Marcel Sabitzer and Konrad Laimer count the Bundesliga side among their former clubs.
Rangnick also enjoyed two stints there as head coach and he admitted playing in Leipzig next week would give his team an advantage.
"Now we have a week to prepare, we won't have to travel a lot, we won't have to fly...
"Apart from the three points we got and the 'home' game in Leipzig, the depth of our squad is better than we thought because the lads who played today did so well."
Sabitzer was the hero at the Olympiastadion against the Netherlands, firing Austria in front for the third and final time in the 81st minute.
"The intensity is the decisive factor, we managed that for a very long time," said the Borussia Dortmund star.
"If you beat the Netherlands, win the group, then you can't be that bad."
