The Barcelona star came to the rescue after the host nation had fallen behind at the Arena de Sao Paulo in Thursday's World Cup opener.
The host nation fell behind to an early own goal from Marcelo, before the tournament's poster boy was perhaps fortunate to be just yellow carded after catching Luka Modric in the throat with a stray arm.
Soon after, the 22-year-old equalized with a fine 25-yard effort on his left foot, before converting a controversial second-half penalty after Dejan Lovren was ruled to have brought down Fred.
And as the game entered injury time, man of the match Oscar sped forward and toe-poked home an effort from the edge of the box to give Brazil breathing space and, most importantly, three points.
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While Brazil had the luxury of naming the same side that started last year's Confederations Cup final triumph over Spain, Croatia was without suspended striker Mario Mandzukic, leaving Nikica Jelavic to lead the line in his absence.
Also without injured left back Danijel Pranjic, Croatia fired the first shot as Mateo Kovacic steered an effort wide in the first minute.
Brazil's first significant foray forward saw David Luiz's tame header saved, but the pressure was on the host at the other end. Ivica Olic's header wide served as a warning for Brazil, and it was one it did not heed, as Croatia claimed a surprise 11th-minute lead.
Olic galloped down the left wing in space before crossing for Jelavic, and although he could only scuff a left-footed shot, his effort was inadvertently steered in by Marcelo from 6 yards.
Brazil's initial response was frantic and anxious, but it soon found its rhythm, with Neymar's run to the byline and subsequent center causing panic, and almost the equalizer, as Oscar saw a 20-yard left-footed effort superbly saved by Pletikosa after Croatia failed to clear its lines.
Neymar's contact with Modric then drew an angry response, with replays showing that the Barcelona star had caught the midfielder in the throat, before Jelavic headed straight at Brazil goalkeeper Julio Cesar.
Croatia's promising start was undone in the 29th minute, though, as Oscar, who did superbly in a cluster of players to find Neymar, watched his teammate trickle a left-footed strike past Pletikosa and in off the inside of the post from 25 yards.
A lull was inevitable following half an hour of mayhem, but it lasted longer than expected. Although Brazil shaded the early stages of the second half, the best it had to show for it was a Dani Alves free kick that sailed over.
That was until the referee pointed to the spot after Oscar's ball to Fred brought about slight contact from Lovren.
Neymar's stuttering run-up was far from convincing, and his tame effort ought to have been kept out by Pletikosa's, but the goalkeeper could only get a hand to the shot on its way into the net.
A sublime Oscar cross nearly brought about Brazil's third goal, only for Luiz to head over, while at the other end Croatia's hard luck was summed up by Ivan Perisic's disallowed goal, ruled out because of Olic's contact with Cesar that appeared minimal.
Modric and Vedran Corluka both went close as well, as Croatia mounted a late charge, but Oscar's composed finish capped a stunning performance from the midfielder, as the host opened the World Cup with victory.