AirAsia CEO Tony Fernandes said he was “devastated” by the news.
President Joko Widodo told media he had instructed all search teams to focus on finding the passengers and crew.
The Airbus A320-200, carrying 162 people from Surabaya in Indonesia to Singapore, disappeared on Sunday.
The head of Indonesia’s search operation, Bambang Soelistyo, says three bodies have been retrieved, not 40 as previously stated by naval officials.
The discovery came on the third day of searching. A navy spokesman said rescuers were “very busy now” with the salvage operation.
The AirAsia statement said the remains were found in the Karimata Strait, south-west of Pangkalan Bun in the Borneo province of Central Kalimantan.
Mr Fernandes said: “I am absolutely devastated.”
He told a news conference there could now be an end to uncertainty for everyone involved.
“This is a scar with me for the rest of my life,” he said.
“It doesn’t change anything. There is at least some closure as opposed to not knowing what’s happened and holding out hope.”
The AirAsia statement said family members would be assigned care providers and an emergency call centre would be set up for those seeking information.
In a news conference shortly after the discovery was confirmed, President Widodo urged relatives to be strong in facing “this difficult moment”.
“I have instructed all the teams to focus on finding the passengers and crew,” he said.
The first debris from the plane was spotted earlier on Tuesday. Pictures of debris and bodies were shown on Indonesian TV. Relatives of passengers on the plane watching the pictures were visibly shocked.
They had been hoping for a miracle, but in the end they had to watch the worst possible news.
Relatives of the passengers screamed and wailed as local television networks showed pictures of what was clearly a human body floating in the water.
Grown men put their hands to their faces. At least two people collapsed and were taken out of the room on stretchers.
The mayor of Surabaya, Tri Rismaharini, went from one crying relative to another, and at one point walked out with a grieving man, while telling him: “We don’t have a choice. Today this happens to you, tomorrow it may happen to me. Nobody knows. So you have to be strong. Our lives belong to God.”
It’s been a trying and exhausting wait for the more than 100 relatives who have been gathering in that room, but no-one could have been prepared for this ending.
The head of the search operation, Mr Soelistyo, said that a shadow was also spotted under the water, which appeared to be in the shape of a plane.
All resources were now being sent to the area where the debris was found, he said.
Mr Soelistyo added that ships with more sophisticated technology were being deployed to check whether larger parts of the plane were submerged beneath the debris.
AFP journalist Bay Ismoyo, who took some of the first photos of the debris, said he saw “an orange object floating on the waters”.
“We saw an unusual object floating. We tried to zoom in and we recognised what looks like a life vest.”
At least 30 ships, 15 aircraft and seven helicopters joined the operation when it resumed at 06:00 local time on Tuesday (23:00 GMT Monday).
The operation, led by Indonesia, includes assistance from Malaysia, Singapore and Australia, with other offers of help from South Korea, Thailand, China and France. The US destroyer USS Sampson is on its way to the zone.
On board the plane were 137 adult passengers, 17 children and one infant, along with two pilots and five crew.
Most were Indonesian but the passengers included one UK national, a Malaysian, a Singaporean and three South Koreans.
AirAsia previously had an excellent safety record and there were no fatal accidents involving its aircraft.