The 22-year-old had been under pressure in the buildup with many Brazilian media calling for him to be replaced by Chelsea teammate Willian.
But he recovered from a nervy start to set up Neymar for Brazil’s first goal, and then completed the win with a brilliant solo strike in injury time.
Oscar, who took time away from Brazil’s preparations to welcome his first child, later told reporters that the only man whose opinion mattered was the manager, Luiz Felipe Scolari.
“The only person who can take me out of the team is Big Phil,” said Oscar, using Scolari’s nickname. “What you (journalists) say doesn’t matter much. What matters is the opinion of the coach.”
After Brazil went behind to an own goal, Oscar’s hard work set up Neymar’s equaliser when he won the ball from two Croatian players in midfield and put the striker into space.
The Chelsea man’s goal was a peach, coming after he ran at the Croatian defence in a solo counter-attack and toe-poked past goalkeeper Stipe Pletikosa.
“To make my debut in the World Cup playing so well, anyone would dream about that, but I have to keep my feet on the ground,” he said.
“We still have two games to play in this round and I hope to play well again in the next one.”
Oscar’s performance was so good that Scolari, perhaps mindful of the criticism he has received, said he deserved the man-of-the-match award as much as two-goal hero Neymar.
But Oscar modestly said there was no doubting who was the team’s shining light.
“I have a good relationship with Neymar, on the field as well as off it. He’s a very cheeky and he has everything to be a star as well,” he said.