Minister Ihor Klymenko said the attack killed the fire chief's wife, daughter and one-year-old grandson.
"On this terrible night, the rescuer was on site with the fire and rescue team to deal with the aftermath of the enemy strikes," he wrote on Telegram, expressing his condolences.
The family was among five people killed when Russia launched six drones to attack the town, regional governor Viacheslav Chaus said.
Six more people were taken to hospital, he added.
The attack came just hours after Donald Trump spoke by phone with Russian President Vladimir Putin.
According to Trump, Putin said "very strongly" that Russia will retaliate for Ukraine's weekend stunning drone attacks on Russian military airfields.
Earlier Ukrainian President Volodymyr Zelenskiy proposed implementing a ceasefire until a meeting can be arranged with Putin.
"My proposal, which I believe our partners can support, is that we agree a ceasefire with the Russians until the leaders meet," Zelenskiy told a briefing in Kyiv.
"At this time, people will understand that the nations, Europe, Ukraine and the whole world have a chance to end the war," he said, adding that monitoring of the ceasefire could be discussed at the meeting.
Zelenskiy said Ukraine would "be grateful" for support for the idea from the US president.
After Ukraine bombed bridges and attacked Russia's fleet of nuclear-capable bombers deep in Siberia and Russia's far north, Putin on Wednesday said he did not think Ukraine's leaders wanted peace.
Shortly after Putin discussed the attacks with top ministers in Moscow, Trump said he had spoken by telephone with Putin for one hour and 15 minutes, and that they had discussed the Ukrainian attacks and Iran.
"We discussed the attack on Russia's docked airplanes, by Ukraine, and also various other attacks that have been taking place by both sides. It was a good conversation but not a conversation that will lead to immediate Peace," Trump said on social media.
Russia has resisted calls from Ukraine and its allies for a ceasefire, saying that certain conditions must first be met.
On Wednesday Putin repeated Russia's stance that any ceasefire would simply be used by Ukraine to acquire more foreign weapons.
Putin said the attacks on the bridge in Bryansk and another one in Kursk had been directed clearly against the civilian population and that the attacks were evidence that the Ukrainian government "was degenerating into a terrorist organisation, and its sponsors are becoming accomplices of terrorists".
"The current Kyiv regime does not need peace at all," Putin said at a televised meeting with senior officials.
"What is there to talk about? How can we negotiate with those who rely on terror?"
Zelenskiy said President Tayyip Erdogan of Turkey, which hosted peace talks on Monday, had expressed support for a top-level meeting of the presidents of Ukraine, Russia, the US and Turkey.
The June 2 talks in Istanbul made little progress towards ending the three-year-old war in Ukraine although the sides exchanged proposals as well as a plan for another major swap of prisoners of war.
Zelenskiy said the POW exchange would begin over the weekend.
"The Russian side has informed us that this weekend, on Saturday and Sunday, it can return 500 people," he said.
Ukrainian Defence Minister Rustem Umerov has previously said the next exchange would focus on swapping the severely wounded and the young, as well as the bodies of dead soldiers.
Zelenskiy's chief of staff Andriy Yermak said on Wednesday he met US Secretary of State Marco Rubio in Washington DC during his visit there.
"We discussed the situation at the frontline and the need to strengthen support for Ukraine in the area of air defence," he wrote on social media.
with AP
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