Real Sociedad's Oyarzabal opened the scoring in the 39th minute, Pedro Porro headed the second in the 66th and Oyarzabal wrapped up matters in the 89th with his fourth goal of the tournament for the European champions.
Marc Cucurella set up both Oyarzabal goals, was also involved in Porro's strike and had a goal chalked off because of a foul in the build-up.
Alex Baena hit a free-kick against the crossbar and star forward Lamine Yamal had a late shot cleared on the goal-line.
Like co-hosts Mexico and title-holders Argentina, who play their last 32 match on Friday, Spain are yet to concede a goal at the tournament in their four games against Cape Verde (0-0), Saudi Arabia (4-0), Uruguay (1-0) and Austria.
Keeper Unai Simon set a World Cup record with now 519 minutes without conceding in a streak which started at the 2022 tournament.
In the last 16 on Monday, the 2010 champions will face Iberian neighbours Portugal with Cristiano Ronaldo or Luka Modric's Croatia, who play later.
"I am happy that I could help the team. It was a complicated match. We knew Austria wouldn't be easy but we had a good day," Oyarzabal said. "We are happy but must continue in this way now."
Coach Luis de la Fuente named La Roja's best performance so far at the event "a wonderful match" and said he was "overjoyed."
Austria, playing their first World Cup knock-out game in 72 years since coming third in 1954, rarely threatened.
Their best chance was a 61st minute header which went over from substitute Sasa Kalajdzic, whose stoppage time equaliser against Algeria had kept them in the tournament.
Austria's Marcel Sabitzer said: "I am disappointed. We wanted to advance. Spain were better overall and deserved winner. We had good moments in the beginning but we were not good enough with the final pass."
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