The clash between two of the form sides of the championship at Mexico City's Olympic stadium looked headed for a goalless stalemate when midfielder Israel Castro put Pumas ahead one minute from time with a shot into the top corner.
Tigres gave away a penalty four minutes into added time when Alberto Acosta brought down Javier Cortes and was sent off and substitute striker Juan Carlos Cacho tucked away the spot-kick.
Pumas are top of Group Three with 28 points from 13 matches, four points ahead of Morelia who won 3-2 at Cruz Azul on Saturday with a last-minute winner from midfielder Elias Hernandez, his second goal of the match.
Cruz Azul are third three points behind Morelia with four matches to go in the qualifying phase of the championship.
The top two teams in each of the three groups plus the next two best-placed teams in the overall standings after the 17-match round-robin phase go through to the quarter-finals.
Tigres lead Group One by two points from title holders Monterrey, who drew 1-1 at home with Santos Laguna in Saturday's clash between the finalists of the Apertura championship in the first half of season.
Santos are a lowly fifth in Group One, nine points adrift of Tigres and seven behind Monterrey.
