A note sent up from the miners 2,300 feet below the surface where they have been stuck since Aug. 5 said: "We are fine in the Refuge. All 33," The Santiago Times reported.
The announcement by Pinera, who was at the San Jose copper and gold mine site, came about 3:15 p.m.
"Coming from the deepest point in this mine, comes a message from our miners who are telling us that they are alive, that they are all together, and that they are waiting to return to see the light of the sun and embrace their families," Pinera said.
The word spread quickly, triggering a round of horn-honking in the streets of Santiago, 500 miles to the south of the mine where the miners are trapped when a main access tunnel collapsed, the Times said.
A sister of one miner told the Times, "We have been dreaming of this moment, but we never thought it would happen."
Local media said multiple notes were attached to the end of a drill rescuers were using to reach the miners. The drill hole also is being used to get food, water and oxygen to the miners.
It wasn't clear how the miners had survived for so long, given estimates their provisions were limited.
Rescue crews are now trying to lower a camera to the miners. Officials said could take up to four months to bring the miners to the surface, the Times said.