Austrian Reichelt and Italian Paris both had the time of one minute and 58.62 for the classic Stelvio course, one of the most demanding pistes in the sport.
World Cup leader Aksel Lund Svindal was 0.01 seconds behind for third place, the Norwegian's sixth top three placing of the winter. Austria's downhill World Cup champion Klaus Kroell was fourth.
It was Paris's maiden World Cup win and Reichelt's sixth but his first in downhill.
"I'm proud to join the list of Austrian downhill winners. It's something that's very important back home. And I picked the right one, the most difficult downhill on the circuit after Kitzbuehel," said Reichelt, the first Austrian winner of a downhill this season.
Paris, a towering skier who has had his technical flaws in the past, said he knew his time would come.
"I didn't ski as well as I did in practice but I gave it my all and in the strong downhill team we have this season, I knew I would have my turn," he told reporters.
It was the 11th tied result in the men's World Cup.
Svindal's third place meant he stretched his lead over defending overall World Cup champion Marcel Hirscher of Austria to 114 points.
"I missed yesterday's practice because I was sick. I was a little bit better today but to be on the podium in these conditions on a piste that never suited me is great, said Svindal.
"Hirscher is so consistent in technical events that every point will count, not only for the big globe but also for the downhill globe, which is becoming one of my goals," he added.
The next men's race is a city event in Munich on January 1, a parallel slalom competition involving the 12 top slalom specialists and the four best overall skiers on the men's circuit.