Two 74th-minute substitutions combined for the winning tally.
Off a throw-in, Sebastian Blanco lofted a ball to Mora, who was making a run toward the net. Mora's head met the ball in stride, and the ball found the net in the final minute of added time.
The Timbers had grabbed an early lead but found themselves playing back on their heels for most of game's remainder.
LAFC finished with a 17-7 edge in shot attempts and a 6-3 advantage in shots on goal.
Timbers captain Diego Valeri scored two minutes into the match. Valeri gathered in a pass from Josecarlos Van Rankin and sent a shot from an angle at the right side of the goal between LAFC goalkeeper Tomas Romero's legs.
It was the veteran's 100th goal across all competitions for Portland, over almost nine seasons.
Carlos Vela scored the equalizer in the 17th minute.
Vela's goal was the result of a pinpoint pass from Diego Rossi into the box, where Vela had made a run before getting his left foot on the ball to put it into the Portland net.
Rossi had a goal off a corner kick disallowed was he was ruled to be offside prior to heading the ball into the net in the 70th minute.
Brian Rodriguez, a designated player for LAFC who was on loan to a second-division team in Spain before playing for his native Uruguay in the Copa America tournament this summer, made his 2021 debut as a substitute in the 79th minute.
The result ended LAFC's three-game winning streak and a seven-game undefeated MLS run against the Timbers, who improved to 5-2-0 at home this season.