Even though it was an inevitable step to prevent contaminated water with higher levels [of radiation] from flowing into the sea, the fact that we had to intentionally release water contaminated with radioactive substances is very regrettable and we are very sorry," said top government spokesman Yukio Edano.
Officials have said that the water being released does not pose a significant threat to human health.
But at a news conference, an official from the plant operator Tepco (Tokyo Electric Power Company) appeared close to tears as he apologised for imposing "another burden" on local residents.
Once the water is discharged, highly radioactive water leaking from the No 2 reactor can be contained in waste storage buildings.
Efforts to stem the leak in a concrete pit at the No 2 reactor with a polymer mix are continuing.
"We tried pouring sawdust, newspaper and concrete mixtures into the side of the pit, but the mixture does not seem to be entering the cracks," said Hidehiko Nishiyama of Japan's Nuclear and Industrial Safety Agency (Nisa).
Tepco said seawater samples taken on 2 April close to the sluice gate of the No 2 reactor contained 7.5 million times the legal limit for radioactive iodine.
It said that the figure had dropped to 5 million by 4 April and that measurements several hundred metres further offshore had fallen to about 1,000 times the legal limit, the Associated Press news agency reported.
Russia says Japan has asked it to send a radiation treatment ship used to dispose of liquid nuclear waste from decommissioned submarines.
The ship, called Suzuran, treats radioactive liquid and stores it. Russia was considering the request, a spokesman for its nuclear agency said

Farm Minister Michihiko Kano says he will increase inspections of marine products because of the leaks, focusing on areas to the south of the nuclear plant.
Elevated levels of radioactive iodine had been found in young launce (a small fish) caught off the coast of Ibaraki prefecture south of Fukushima, Kyodo news agency reported, citing the health ministry.
Levels of 4,080 becquerels per kg - a measurement of radioactivity - had been detected, the ministry said. The limit for vegetables is 2,000 becquerels per kg - officials said there was no fixed limit for fish but they would seek to apply the same limit.
Tepco, meanwhile, says it will begin paying money to residents and farmers who live and work around the plant by the end of this month.
Some 80,000 residents have had to evacuate, while restrictions on sales have hit farmers.
Tepco has already begun paying money to local governments to help evacuees from the plant exclusion zone.
On Tuesday, shares in the power giant hit a record low of 362 yen (£2.65) amid concern over the Fukushima plant.
Across Japan, more than 161,000 people from quake-ravaged areas are living in evacuation centres, officials say.
The official death toll from the 9.0-magnitude earthquake and tsunami which struck on 11 March stands at 12,344, with more than 15,000 people still unaccounted for.
More than 80% of the victims have been identified and their bodies returned to their families.