"The only thing we could be accused of is that we didn't score enough goals from the chances we had, and that we didn't work well twice for set pieces," Rangnick said.
The game got off to a ferocious start. Both teams had chances inside the first 30 seconds and Demiral scored inside the first minute after the Austrian defense failed to deal with a corner.
Baumgartner's attempted clearance rebounded back off teammate Stefan Posch to goalkeeper Patrick Pentz, who scooped it off the line only for Demiral to pounce on the rebound and score.
UEFA clocked the goal at 57 seconds, making it the second-fastest ever at the Euros. Albania scored after 23 seconds against Italy in the group stage.
Austria almost responded but Demiral somehow did enough to keep Baumgartner from scoring on the line.
The Austrians pushed hard for an equalizer but the Turkish players matched their intensity and might have scored more — Turkey star Arda Guler tried his luck from distance and Demiral went close from corner.
Rangnick responded at the break by sending on forward Michael Gregoritsch to partner Marko Arnautovic, who was then denied in a one-on-one by Gunok.
Austria's corner-taker Marcel Sabitzer was also pelted with beer cups at the other end and at one point went down on the ground clutching his head to indicate he had been hit by an object, but was quickly back on his feet.
Austria's goal also came from a corner as Stefan Posch headed on a delivery for the unmarked Gregoritsch to volley home at the back post.
Neither team could add any more despite a flurry of chances in the rain, with Gunok coming up with his biggest save in the final seconds.
Both teams were without their captains. Turkey's Hakan Calhanoglu was suspended and Austria's David Alaba, arguably the country's best player, is still recovering from a knee injury that kept him from playing in the tournament.,
Alaba was there to support the side as it surprisingly topped a group with France and the Netherlands, and he shared a warm embrace with Guler, his Real Madrid teammate, before kickoff.
Cody Gakpo broke a scoreless tie in the 20th minute and Donyell Malen scored two late goals to send the Dutch to their first quarterfinal in the tournament since 2008.
The Dutch had missed a plethora of earlier opportunities to add to the lead and captain Virgil van Dijk also hit the woodwork. It was nevertheless a much-improved display compared to its disappointing group stage performance, which ended in a 3-2 loss to Austria.
"I think we played a good game and showed a great reaction from the last one," Gakpo said. "Especially after the last game we needed a reaction and today was a good step in the right direction. … "We talked a lot the aggression, the intensity and defending like a team."
It was Gakpo's third goal of the tournament, making him joint top scorer along with Germany's Jamal Musiala, Georges Mikautadze of Georgia and Ivan Schranz of Slovakia.
Romania had dominated possession until Gakpo's goal, but never really tested Netherlands goalkeeper Bart Verbruggen. Instead the opener came when Xavi Simons surged forward before finding Gakpo on the left. The Liverpool forward cut inside the blue-haired Andrei Rațiu and fired into the near bottom corner.
The goal seemed to give the Dutch confidence and they had several chances to extend their lead. A huge section of their orange-clad fans thought they scored again six minutes later when Stefan De Vrij was allowed a free header on a corner but he planted it just wide of the right post, rippling the sidenetting.
A goal-saving tackle from Radu Dragusin denied Memphis Depay a tap-in, while the Netherlands went even closer to doubling its lead on the stroke of halftime.
Denzel Dumfries did well to keep the ball from crossing the byline and cut it back to a completely unmarked Simons, who perhaps should have hit it first time but instead took a touch allowing the Romania defenders to close him down and, under pressure, he fired wide of the right post.
Romania came very close to putting the ball in its own net when Bogdan Racovițan cleared Depay's effort off the line and it came off teammate Marius Marin, but the ball fell into the arms of goalkeeper Florin Niță.
Van Dijk also saw a header come off the right post.
Gakpo had a second-half goal chalked off for offside, but turned provider seven minutes from time as he managed to just keep the ball in play, under pressure from Dragusin, and proded it back from the byline for Malen to poke home.
And the Dutch capped a dominant performance in stoppage time when Simons released Malen for a surging run from the halfway line before slotting past Niță.
With files from Daniella Matar, The
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