Brazil, who had previously never lost a World Cup qualifying game on home soil, were reduced to 10 men when Newcastle United’s Joelinton was dismissed on 81 minutes.
The match between South America’s biggest rivals, was delayed for 30 minutes to crowd trouble ahead of kick-off.
Fighting broke out in the stands and Lionel Messi tried to calm the situation down before the teams were taken back to the dressing room until order was restored.
When play did begin, it proved a tempestuous, fractious encounter.
Alvarez gave a typically industrious performance alongside Messi, and it was a towering header from former City defender Nicolas Otamendi that settled the game on 63 minutes.
Brazil have now lost three qualifiers in succession and lie in sixth in the table - eight points behind world champions Argentina.
However, with 12 games remaining and the top six qualifying, Brazil are highly likely to still make the cut.
Elsewhere, Josko Gvardiol helped Croatia secure their place at the 2024 European Championships in Germany.
The defender started as Croatia beat Armenia 1-0 in Zagreb to guarantee a top-two finish in Group D.
Needing to win to be certain of a second-placed finish, Ante Budimir’s 43rd minute strike was enough to clinch victory as they finished clear of Wales who could only draw with top-of-the-table Turkey.
It is the seventh successive tournament that Croatia have qualified for, with Gvardiol part of the side that finished third at last year’s World Cup in Qatar.
He will joined by team-mate Manuel Akanji despite Switzerland’s defeat away to Romania.
The Swiss finished second in Group I but missed the chance to finish the group on top following a 1-0 loss in Bucharest.
Romania’s Denis Alibec scored the only goal in the 50th minute with both countries already on their way to Euro 2024.
Ederson and Julian Alvarez will be on duty later for a World Cup qualifier at the Maracana Stadium in Rio de Janeiro.
Brazil face Argentina at 00:30 (UK) on the back of two defeats while the World Cup winners currently sit top of the table.
