Tigres took the lead in the seventh minute when left winger Alvarez raced onto a pass from Chilean striker Hector Mancilla, beating two defenders to shoot left-footed past goalkeeper Oswaldo Sanchez from a narrow angle.
Santos were down to 10 men for more than an hour after Juan Pablo Rodriguez was sent off midway through the first half for a bad tackle on fellow midfielder Manuel Viniegra.
The Tigres defence, best in the league phase of the championship when they conceded 13 goals in 17 matches, coped well with the home side's attack though they had a couple of scares.
Santos striker Oribe Peralta hit the bar with a shot from outside the box in the first half and five minutes from time Juan Cardenas crossed low from the left and Peralta lunged forward but just failed to make contact.
The sides meet again at the Estadio Universitario in the northern city of Monterrey in the second leg on Sunday when Tigres will be looking for their third league title and first in 29 years.
Santos, who have won the title three times since the 1996/97 season, are in their fourth final in two years but lost the two they played in 2010.
The Mexican season has been divided into two championships, Apertura and Clausura, since 1996/97.
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