Brazil forward Robinho, making his first start for the club since his January return on a six-month loan from Manchester City, had two scoring chances but goalkeeper Sidney saved well.
Rio Claro, 19th in the 20-team (Sao Paulo state) championship, took a surprise lead five minutes before half-time through Jackson and Santos went off at the interval to boos from their fans.
The leaders equalised in the 70th minute when 18-year-old Neymar, known as the new Robinho, set up Andre to tap the ball home.
Neymar also created the 89th-minute winner when he dribbled past two defenders and had his shot blocked by Sidney but 38-year-old former Barcelona midfielder Giovanni, who had come on as a substitute, headed in the rebound.
With the carnival this weekend, including the expected appearance at the Rio de Janeiro Sambadrome of top players Adriano, Ronaldo and Robinho, Santos's match was the only one scheduled on Sunday in the three main state championships that open the season.
In the Carioca (Rio) championship, Vasco da Gama beat Fluminense on penalties after a 0-0 draw at the Maracana on Saturday to reach the final of the first stage.
Vasco will meet the winners of Wednesday's second semi-final between holders Flamengo and Botafogo.
In the Gaucho (Rio Grande do Sul) championship, Saturday's matches went ahead at 6pm GMT after referees verified the temperature was below 35 degrees Celsius as decreed by a Porto Alegre judge in a recent ruling to protect players in an extremely hot summer.
The temperature was 32.5 degrees in the shade at kick-off time for Gremio's 2-1 home win over Sao Jose and 32 degrees for Internacional's victory by the same score at Esportivo.
Inter and Gremio went through to the second phase in first and second place.
