Croatia stunned a fervent Sao Paulo crowd as Marcelo scored an own goal to give them a deserved 11th-minute lead in the Group A clash.
Oscar was central to his team's revival, laying on Neymar's equaliser and scoring a superb third after the Barcelona forward netted a controversial second-half penalty.
But the Chelsea playmaker was not about to bask in the glory of victory.
He told Globo: "Today we knew it would be difficult. We (gave away) the lead too early, we have to have more attention and get it right for the next game."
Having modestly claimed he only elected to shoot successfully from distance in the closing stages due to cramp in his legs, Oscar dedicated the strike to his newborn daughter.
He added: "My daughter was just born. (The goal is) not only for her, but for my wife and family."
Oscar's former Chelsea team-mate David Luiz suggested the scoreline might have flattered the hosts, who found themselves on the right side of some questionable calls from the match officials.
"No doubt, I think the score does not depict what the game was," Luiz told Globo.
"But Croatia has a very skilled team, great players, experienced.
"Our friendlies have (prepared us) for this type of game.
"It was the first step, now (the aim) is to work hard, with great vigour, knowing that other games will not be easy."
