The strike occurred on Tuesday in the city of Poltava, the capital of the region of the same name, officials said.
Poltava is located about 110km from the border with Russia and about 350km southeast of Kyiv.
The strike appeared to be one of the deadliest carried out by Russian forces since the war began more than 900 days ago on February 24, 2022.
"One of the buildings of the Institute of Communications was partially destroyed. People found themselves under the rubble. Many were saved," Zelenskyy said in a video posted on his Telegram channel.
I received preliminary reports on the Russian strike in Poltava. According to available information, two ballistic missiles hit the area. They targeted an educational institution and a nearby hospital, partially destroying one of the telecommunications institute's buildings.… — Volodymyr Zelenskyy / Володимир Зеленський (@ZelenskyyUa) pic.twitter.com/TNppPr1OwFSeptember 3, 2024
"All necessary services are involved in the rescue operation," he said.
He said he had ordered "a full and prompt investigation" into what happened.
Zelenskiy did not provide further details.
The missiles hit shortly after the air raid alert sounded, when many people were on their way to a bomb shelter, Ukraine's defence ministry said, describing the strike as "barbaric".
Rescue crews and medics saved 25 people, 11 of them dug out from the rubble, a defence ministry statement said.
Zelenskiy repeated his appeal for Ukraine's Western partners to ensure swift delivery of military aid.
He has previously chided the US and European countries for being slow to make good on their pledges of help.
He also wants them to ease restrictions on what Ukraine can target on Russian soil with the weapons they provide.
Some countries fear that hitting Russia could escalate the war.
"Ukraine needs air defence systems and missiles now, not sitting in storage," Zelenskiy wrote in English on Telegram.
"Long-range strikes that can protect us from Russian terror are needed now, not later.
"Every day of delay, unfortunately, means more lost lives."