The league leaders have dropped just two points from their opening seven top-flight games - a 1-1 draw at Leon on the opening day - but were made to work hard for their latest victory by a determined Monterrey side.
Jose Guadalupe Cruz's team looked to have done enough to earn a valuable draw at the Estadio Azteca, but Luis Rey's 88th-minute penalty was enough to give the reigning Clausura champions all three points.
Morelia are now the closest challengers to Miguel Herrera's pacesetters, even though they could only manage a 2-2 draw at bottom-placed Atlante on Sunday.
Ruben Israel's winless Atlante side twice took the lead against their visitors, but were pegged back by equalisers from Hector Mancilla and Joel Huiqui.
Despite Morelia's failure to win in Cancun, they still moved up to second in the standings by virtue of Leon's first defeat of the season, which came thanks to Carlos Ochoa's first-half strike for Chiapas.
And Cruz Azul are now level on points with Gustavo Matosas' side after they came from behind to snatch a 2-1 victory at Toluca.
Juan Cacho opened the scoring for the hosts after just two minutes at the Estadio Nemesio Diez, but second-half strikes from Mariano Pavone and Alejandro Castro gave the visitors the win.
Queretaro are fifth after recording their third consecutive top-flight win with a 1-0 home victory over Puebla on Saturday. Diego De La Torre grabbed the all-important goal for the hosts shortly after the interval.
At the other end of the table, second-bottom Pumas UNAM earned a creditable 0-0 at Pachuca, while Guadalajara were 3-0 to the good at Tigres UNAL with just 14 minutes remaining but threw away two points by being hauled back to 3-3 through late goals from Alan Pulido, Juninho and Manuel Viniegra.
Elsewhere, Tijuana beat Veracruz 3-0 at the Estadio Caliente and Atlas drew 2-2 at home with Santos Laguna.
