The defeat, however, left Pumas still four points clear of Cruz Azul, who lost 3-0 at Santos Laguna, and Morelia, who play on Sunday at home to Pachuca, in Group Three after 12 matches.
Salazar scored in the 19th minute to open the scoring for Chiapas, who are bottom of the group after only their third win, then increased the Jaguars' lead when he scored from Damian Manso's pass.
Edgar Andrade made the score 3-0 in the 33rd minute before substitute Fernando Espinoza pulled one back with a 79th minute header as Chiapas sat back and defended in the second half.
The top two teams in each of the three groups and the next two best-placed teams in the overall standings at the end of the 17-match round-robin phase qualify for the quarter-finals.
San Luis lost 3-2 at Queretaro but remained top equal in Group Two with Atlante, who lost 2-0 away to Group One leaders Tigres UANL, and America, who are at home to Puebla on Sunday.
Former Uruguay forward Carlos Bueno scored twice as Queretaro recovered from going two goals down in the opening half hour.
Santos Laguna, runners-up to Monterrey in the Apertura championship in the first half of the season, climbed off the bottom of Group One with their first win under coach Diego Cocca after a run of five defeats.
Striker Darwin Quintero shocked Cruz Azul by giving Santos the lead in the second minute with a close in header from a corner and while Cruz Azul dominated play, defender Ivan Estrada slipped through the defence to add the second on the half hour.
Midfielder Juan Pablo Rodriguez rounded off the victory with a 72nd-minute penalty for a foul on Quintero, who was sent off for a head-butt after retaliating for the tackle that led to the penalty.
