Hungary coach Bernd Storck said Austria were "obvious" favourites to win the teams' Euro 2016 clash and that the 2-0 victory was a dream come true.
Austria, whose squad includes David Alaba and Marko Arnautovic, have been tipped to impress this summer while Hungary had been expected to face a tough battle to get out of Group F.
Marcel Koller's side came close to an early lead when Bayern Munich star Alaba struck the post, but they fell behind to an Adam Szalai strike in the 62nd minute -- his first goal in 41 matches -- and Austria defender Aleksandar Dragovic was shown a second yellow card soon afterwards.
Hungary sealed victory through Zoltan Stieber's neat chip in the 87th minute to make the victory certain.
Storck said after the match: "When you look at the Austria team and compare it to ours -- the experience, the champions they have -- it was obvious who the favourite was. My team has developed well, including in recent friendlies, and we got our reward.
"We knew what to expect from Austria and we knew what their strengths were. We took their important players out of the game which was key, and that was vital to our success."
He added: "You can see how the fans reacted -- that was brilliant. A dream has come true. We hoped for it, we wished for it...
"When everyone does their hard work then it pays off. A huge compliment to everyone who helped make sure that we could decide this game."
The match also saw goalkeeper Gabor Kiraly become the European Championship's oldest ever player at 40 years and 75 days and Storck said: "Kiraly has been rewarded for the years of hard work he has put in."
The match turned on Szalai's goal but the Hoffenheim striker, who spent the end of the season on loan at Hannover, did not want to draw attention to his efforts.
"I don't want to talk about myself. The whole team won," Szalai said. "Everyone worked together. I think we managed to do everything our manager told us in terms of advice and strategy."
Hungary forward Tamas Priskin said they would now look to secure their place in the next round against Iceland on Saturday.
He told reporters after the game: "It was a great night for us. I am very happy we won the game and scored two nice goals, we have to continue on this way on Saturday.
"It was very special to be here, but it is more special now we have won a game. Everyone is very happy but we have to be concentrated on Saturday, and hopefully we can win that and qualify.
"Everybody was working really hard in the last few weeks and today that has paid off for the team. We did very well at the back and in attack. I am very happy."
Fellow forward Krisztian Nemeth added: "We have worked so hard for this, and we did it. It was a good result for the Euros and maybe a surprise for the fans. Nobody expected we would win our first game. The team believed. In the end, we deserved to win.
"We have worked a lot in the few weeks we have had together, we have done very well, and we have all the information for what we need for this game. We tried to qualify for the next round, that is the target from the beginning, and now we are closer. Not a bad start."
Nemeth praised the supporters' contribution, saying: "It was goosebumps when they were singing the anthem. It is an amazing memory for us and the fans. I have been told the whole country is celebrating."
ESPN FC's John Brewin contributed to this report.