Zico turned Italian defender Claudio Gentile before feeding Socrates, who knocked it past keeper Dino Zoff for the equaliser.
But on 25 minutes a mistake by Toninho Cerezo allowed Rossi to nip away and score his second for 2-1.
Brazil pressed hard and Zoff, the 40 year-old Italian captain, pulled off a string of saves, notably from Cerezo.
"I always felt that Zoff at that time was over his best," said former Tottenham star Garth Crooks.
"I was probably listening to too many commentators, because everyone assumed that, because he was 40, he was past his best. When you look back, he was probably at his best."
But Brazil finally broke through and equalised for a second time through Falcao.
The scoreline would have suited Brazil. But their relief was short-lived. A shot by Marco Tardelli was turned in by Rossi for his hat-trick.
And still the drama continued. Both Socrates and Giancarlo Antognoni put the ball in the back of the net for their respective teams - neither goal was given.