Police removed four bodies from the Mermaid tourist boat, AFP reported.
Another four people are still missing some two weeks after the tragedy, the worst such incident in Hungary in 75 years.
The operation to lift the vessel out of the river onto a waiting barge began at around 6:00 a.m. (0400 GMT) and took more than six hours to complete.
Several dozen rescue workers took part in the operation, including members of a diving team sent from South Korea.
The Mermaid was carrying mostly South Korean holidaymakers when it capsized and sank on the evening of May 29, seconds after colliding with a bigger river cruise boat on a busy stretch of the river.
Those on board included a six-year-old girl travelling with her mother and grandparents, and the Mermaid's Hungarian captain.
Early on in the operation, divers removed four bodies from the vessel.
"It is presumed that the only child victim and the captain are two of the four victims recovered today," a police spokesman Kristof Gal told reporters later.
"The identification process is still ongoing," said Gal.
Only seven of the 35 people on board are known to have survived the accident, with the prospect of finding any more passengers alive considered to be practically zero.
The Mermaid will be transported to a dock in southern Budapest later Tuesday, said police.
"Four people are still missing, and while it is still possible that further bodies will be found on the boat on later examination, the forces deployed in the search operation of the southern sections of the Danube have been doubled," Gal told AFP.
Several of the victims found before Tuesday were discovered along the Danube, dozens of kilometers south of the Hungarian capital.
Strong river currents
Since the accident, divers had been unable to enter the submerged boat due to the strong current in the river, which is swollen from weeks of rain.
The salvage crew instead focused on fixing wire harnesses underneath the vessel to prepare it for hoisting by a powerful crane mounted on a barge.
Hungarian police, who led the team of divers and experts, initially estimated the lifting would take around six hours, but work was delayed several times.
Temperatures soaring above 30 degrees Celsius (86 degrees Fahrenheit) and strong river currents slowed the divers' work, while an extra wire rope was needed to support the damaged part of the Mermaid after it surfaced.
"This rescue operation was greatly taxing physically and mentally," Peter Kolozsi, a spokesperson for emergency services members of the salvage team, told a Hungarian public media news program.
During the operation, divers periodically entered the vessel to search for bodies of any victims still inside.
At one point, a member of the rescue crew fell into the fast-flowing water and had to be rescued after almost being swept away.
On Monday the river cruise boat, the Viking Sigyn, was again searched by Hungarian police who have launched a criminal probe into the cause of the accident.
After the incident, the boat had travelled on to Germany before returning to Hungary according to its schedule.
The investigation, at a river port in Visegrad 50 kilometers (30 miles) north of Budapest, was to "acquire more evidence... and to gain wider context of the accident", said Gal.
Its 64-year-old Ukrainian captain was arrested by Hungarian police days after the incident on suspicion of "endangering waterborne traffic resulting in multiple deaths".
According to Hungary's association of passenger shipping companies, the sinking of the Mermaid is the first such accident to cause mass fatalities in Hungary for 75 years.
For many in South Korea the tragedy has brought back memories of the 2014 Sewol ferry sinking, which claimed more than 300 lives in one of the deadliest maritime disasters in the country.
The Mermaid accident happened on a popular part of the Danube River for pleasure trips, from where passengers can view the city and parliament building illuminated at night.
Dozens of small sightseeing boats ply the river through Budapest every day.
Larger river cruise boats travelling on the Danube between Germany and the Black Sea typically spend several days moored in the capital.