"I had a good night. So far, I'm fine, I'm happy with my feelings," said the Belgian, one of the race favourites, before stage six got underway in Naples.
Evenepoel's first spill came when he was brought down by teammate Davide Ballerini (28) who wobbled and fell after a dog ran out onto the course.
The second crash came less than three kilometres from the finish.
According to Toon Cruyt, the doctor for his Soudal-Quick Step team, he has a "muscle hematoma" and was hurt in his sacrum (at the base of the spine).
"It's not nice, especially early in the Giro like this," said Evenepoel.
"(I have) some pain in the back, some big bruises. I've got a big black spot on my back which is some blood.
"It's life, it's racing. We have to deal with it and just handle the pain as well as possible and try to relax as much as possible because I think I will need a lot of energy now to heel from the wounds.
"Up to now, everything feels pretty well, I'm happy with the feeling."
Thursday's sixth stage, a 162-kilometre loop around Naples, ought not to present any major difficulties but Friday's stage, which includes a final climb above 2,000 metres in altitude, promises to be tougher.
"I just hope I don't have a bad day tomorrow, we'll see," said Evenepoel. "It's good that today is a hard day that my legs will be pushing and turning."
