A second half goal from Paraguayan Carlos Gonzalez had handed home side Pumas the lead, with Leon center-back Jaine Barreiro receiving his second yellow card and getting sent off with nine minutes to go.
But against the run of play and with Pumas searching for a second goal, David Ramirez crossed for Gigliotti to level the scores in the 89th minute.
The defining second leg will be played in Leon on Sunday. Both games are being played without fans in the stadiums due to the COVID-19 situation in Mexico.
"It was a very strategic game, them defending their style and us ours," said Leon coach Ignacio "Nacho" Ambriz in a video press conference. "It was a very even game, with a lot of errors in moving around the ball. The draw is fair."
"We have to play well at home [in the second leg], take the game to them like we do and take advantage of being at home, where we haven't lost [this season]."
Leon went into the game in Mexico City's Estadio Olimpico Universitario as the slight favorite, but Pumas had won 4-0 in an epic semifinal second leg against Cruz Azul last Sunday and took the initiative early in the first half.
Juan Iturbe spawned two chances to put Pumas into the lead before half-time, with Chilean Jean Meneses having Leon's best opportunity with the score 0-0 at the break.
The match had been fairly even, but the deadlock was broken when a perfectly-weighted right-wing cross from Carlos Gutierrez found Gonzalez free to head past Rodolfo Cota in the 72nd minute, leaving everything to play for in the second leg.
Leon finished the regular season in first position, with Pumas in second. Both teams only lost one game in the regular season.
Monterrey won the 2019 Apertura one year ago and is still the reigning Liga MX champion, after the 2020 Clausura was suspended and eventually canceled due to the COVID-19 pandemic.