Matt timed 52.89 seconds on the sun-soaked Podkoren course to lead Germany's Felix Neureuther by 0.05.
Henrik Kristoffersen of Norway, who won the junior world title in the discipline four days ago, was third, 0.30 behind Matt.
"It was a clean run," Matt said. "I could have gained more time in the upper section but the bottom part was really good. I have to charge again in the second run, it's all very tight."
Neureuther led Matt at the first split but lost time in the steep middle section.
"There were no huge mistakes but a couple of turns were not perfect," Neureuther said. "A solid run, that was my goal. It won't get easier in the second leg."
Two-time defending overall champion Hirscher had 0.56 to make up in the second run later Sunday.
"I don't understand the deficit," Hirscher said. "The feeling was completely different but fact is I am six-tenths behind. I am just not racing as easily as I usually do."
Hirscher trails overall World Cup leader Aksel Lund Svindal by 41 points and needs to finish fifth or better to overtake the Norwegian, who doesn't compete in slalom.
In an unlikely scenario, Hirscher could lock up the slalom title Sunday.
Ted Ligety, who won a record sixth giant slalom on the same course Saturday, was 1.42 back in 16th. Former overall champion Ivica Kostelic of Croatia, who won here last year, finished 2.23 off the lead in 24th.
Sunday's race is the last event before the World Cup finals start in Lenzerheide, Switzerland, on Wednesday.