Uruguayan Gonzalo Vargas scored for Atlas in the third minute of Saturday's game and Bruno Marioni missed the chance to extend their lead when he had a 35th-minute penalty saved by Federico Vilar.
It was Atlas's first win in five matches in the tournament.
La Volpe, one of Latin America's most outspoken coaches, took over at the Guadalajara-based club at the end of January.
He has previously coached Boca Juniors and Velez Sarsfield in his native Argentina and Mexican side Monterrey since leading Mexico to the second round of the 2006 World Cup.
Big-spending America held Chiapas to a 0-0 draw away from home in their first game under Jesus Ramirez, their fifth coach in a year.
Ramirez, coach of the Mexican team which won the under-17 world championship, replaced Ramon Diaz last week.
America, who have spent around $20 million on new players in the mid-season break, stayed bottom of Group Two with five points from five games. Chiapas lead Group Three with 10 points.
Unbeaten Pachuca continued their impressive start by beating Pumas UNAM 3-2, Paul Aguilar scoring the winner in the 86th minute. The Beavers top Group One with 13 points from five games.
The 18 teams meet each other once in the qualifying stage but are divided into three groups of six. The Clausura is the second of two championships played in the Mexican season.
