Six-times Brazilian champions Sao Paulo fell apart after Luan was left unmarked to side-foot Cruzeiro ahead five minutes into the second half.
On loan to Cruzeiro after playing for relegated Palmeiras last season, Luan added two more goals in a three-minute spell late in the game after catching the home defence off guard.
Paulo Autuori, who led Sao Paulo to the South American Libertadores Cup title in 2005, was brought back for a second stint as their coach two weeks ago and has seen his new side lose three times in a nightmare eight-day spell.
Sao Paulo, usually a dominant force in Brazilian football, dropped to 15th in the 20-team table with eight points from nine games and are already considered relegation candidates.
They have lost four league and two cup games during their six-game skid.
Gremio finished with nine men in a 2-1 loss at Criciuma in another of Saturday night's games.
Matheus Biteco was dismissed in the 23rd minute for lashing out at an opposition player who tried to tackle him and forward Eduardo Vargas, on loan from Napoli, was dismissed early in the second half for kicking an opponent.
Wellington headed Criciuma in front two minutes after the first red card, before midfielder Ze Roberto, 39, levelled for Gremio before halftime.
Criciuma made the most of their two-man advantage when Matheus Ferraz turned in Sueliton's cross in the 75th minute.
Two second-half goals from Elias and Renato gave Botafogo a 2-0 win over Nautico that left them top of the table, though the match was played in front of a disappointing 5,000 crowd.
The game was briefly interrupted when a dog walked onto the pitch and calmly stood on the edge of the penalty area before being carried off by a Nautico player.