It seemed apt that after his Austria side ignited their hopes of reaching the last 16, Ralf Rangnick – famed for imposing a high-pressing style across the Red Bull group – plumped for an energy drink, pouring himself a glass in his post-match press conference. A well-earned victory provides hope of qualifying from Group D when they meet the Netherlands here on Tuesday.

For Rangnick it was time to refuel after his side overpowered Poland, whose tournament is over on the back of the goalless draw between Netherlands and France. Marko Arnautovic, emboldened with the captain’s armband, crowned victory from the penalty spot but after Christoph Baumgartner restored Austria’s advantage, he hurtled towards Rangnick in celebration to give him a bear hug.

“My glasses are still OK,” Rangnick said. “I said before the match that it would be a playoff match. The team that loses is out. The team who wins has everything in their own hands. Since I’ve been Austria coach, it was the most important match. That is why we are so content.”

From a Polish perspective, this game did not pan out as expected. Firstly Robert Lewandowski did not start, instead introduced from the bench on the hour despite his manager, Michal Probierz, confirming the striker was fully fit, and frankly they appeared strangely startled at Austria’s intensity. Had they not finished their homework on Rangnick’s philosophy? Perhaps Austria were buoyed by the initial absence of Lewandowski, who later entered to a hero’s welcome.

It seemed inevitable Austria would soon have something to show for their fast start and took the lead when Gernot Trauner headed in Philipp Mwene’s cross on nine minutes. Rangnick tried his best to keep his emotions in check but could not resist a quick kerching, clenching his left fist in celebration like a man who had just won £50 on the premium bonds. Krzysztof Piatek replied after half an hour, feasting on a loose ball inside the box but Austria rarely lost control.

This game was supposed to be the moment when Lewandowski signalled his arrival but his stay at this tournament is set to be an unhappy footnote. Instead it was Austria who had lift-off and another substitute who made the difference three minutes after entering. Alexander Prass skipped down the left and then passed infield, his low diagonal ball seemingly for Arnautovic. Or so Poland wrongly presumed. Arnautovic, an extrovert who relishes the role of showman – either that or pantomime villain – allowed the ball to whistle between his legs, magically sliding open the doors of the Poland defence. Baumgartner looked up, picked his spot and curled past a wrong-footed Wojciech Szczesny.

Austria, beaten by France in their opener, have qualification in their own hands. “We won against Germany in one of our friendlies last year and we have had a really good period in the last few months,” Baumgartner said. “People start to think big quite quickly but as a team we knew how to assess the situation. We know if we reach our maximum we can beat a lot of nations.”

Arnautovic capped victory from the spot after Szczesny upended Marcel Sabitzer as the forward rounded the Poland goalkeeper. Szczesny said Lewandowski’s presence would spook Austria but the impact of the 35-year-old Barcelona striker was minimal at best, 11 forgettable touches. Lewandowski replaced Adam Buksa, who scored against the Dutch last Sunday. “We conceded a goal very soon after he came on, so things changed,” Probierz said. “We knew this would be a highly intensive match and we decided with the medical team and Robert that he would not start. We hoped that we could break through in the second half.”

Before arriving on to the pitch, Lewandowski’s greatest intervention was to jump out of his seat in the far left of Poland’s XXL dugout midway through the first half and whisper words of wisdom to a member of backroom staff, who then relayed the message to another coach. Whatever Lewandowski said, it seemed to have the desired effect, Piatek side-footing in minutes later.

It was a short-lived moment of joy in a one-sided game. Baumgartner instantly made a beeline for Rangnick after scoring and soon afterwards Szczesny saved from Patrick Wimmer as Austria went for the jugular. Arnautovic ensured a nerveless crescendo, sending the goalkeeper the wrong way from the spot.

By the end Austria were running amok, Konrad Laimer going close to finding a fourth. “There was a lot of pressure,” Baumgartner said. “We felt it yesterday and we couldn’t even have a good afternoon nap because there was so much stress, we felt so much pressure on us, so we are really relieved. We still know we have a lot of work in front of us. We are only just starting.”