With only two minutes left in regulation, Morales broke loose to receive a long pass moments after Stansbury (12-4-3) tried to break the 2-2 deadlock with a corner kick play.
After beating one last defender, Morales chipped in the game winner to trigger a jubilant reaction from the Flyer bench and supporters.
"It was exciting, coming back from two-one," said Morales, who admitted to being "really nervous" after the Stallions took a 2-1 lead on a Kaden Littlefield goal in the 63rd minute.
"It was a counterattack and I cut [in front of] him and crossed," said Morales of the winning goal. "Then I don't know what the kid did."
"We were able to slip that one defender, we made the pass out and Tauri is very creative," Dixie coach Burton Myers said.
The Dixie forward also tied the game five minutes earlier when, after the Stansbury keeper came out of the net area, he was able to poke the ball into the open net.
Dixie also started the game very well, getting four corner kicks and a 25-yard direct kick in the first six minutes, although the visitors were dismayed at the field conditions.
"This is a state playoff game and you come up to a 55-by-110-yard field where the grass is five inches long and they've got a beautiful field right across the way," said Myers, pointing over to the direction of the Stansbury football field. "I just don't understand it."
In the 27th minute, lightning in the area forced a delay in play. While the Stallions went to a nearby church, Dixie's team went back onto the bus to wait out the 53-minute delay.
Coming back to the field, Stansbury became the first to score as Caleb Andersen on a pass from Brandon Bustillos found the net.
Seven minutes into the second half, a rush on the right sideline by Tyler Bennett led to Morales' first goal as Dixie tied the game at 1-1.
In the state semifinals next Friday, at Alta High School, Dixie will play against Juan Diego. The winner will advance to the Class 3A state championship on Saturday.
"It tells what kind of team we've got, to come through this adversity," Myers said. "To come and play on a pitch like this and to come away with a victory with the rain and the lightning we stayed together and sucked it up the last 10 minutes."