Kleber's run began when he struck the outside of the left-hand post with a bicycle kick in the 21st minute, then hit the foot of the other post one minute later.
In the 31st minute, he struck the right post again and three minutes later burst down the left and beat his marker only to foiled by the woodwork for a fourth time.
In between, Ricardo Xavier fired Ituano ahead in the Paulista championship match.
Two other Santos players also struck the post while an injury-time goal from Alex Afonso handed Pele's former team their first defeat of the season.
Sao Paulo lost by the same score at home to Santo Andre to end the Brazilian champions' 22-match unbeaten run.
Osny, left completely unmarked, and Junior Dutra scored early in each half while Sao Paulo had defender Miranda sent off on the hour for a bad tackle on Osny.
Sao Paulo's last defeat was against Gremio in August in the Brazilian championship.
Palmeiras striker Lenny scored a hat-trick, including an 88th minute penalty, to help Vanderlei Luxemburgo's team to a 3-2 win at Ponte Preta which maintained their perfect start to the season.
Palmeiras lead the Paulista with 12 points from four games and also beat Bolivia's Real Potosi 5-1 in a Libertadores Cup tie on Thursday.
The Paulista is the biggest of the regional championships which kick off the Brazilian season.
