In a statement published on the Telegram messaging app, Vyacheslav Gladkov said two industrial facilities in the border town of Shebekino had been shelled and four employees had been wounded.
Several settlements were left without electricity, he added.
Belgorod, which borders Ukraine's Kharkiv region, has repeatedly come under attack from Kyiv's forces since the beginning of the full-scale conflict in Ukraine in Feb. 2022.
Last week, Ukrainian forces undertook a major cross-border raid on a series of border villages, briefly seizing several settlements before withdrawing over the frontier.
Kyiv almost never publicly claims responsibility for attacks inside Russia and on Russian-controlled territory in Ukraine, but has said that destroying infrastructure is preparation for a planned ground assault.
Russian President Vladimir Putin on Sunday ordered stronger border security to ensure "fast" Russian military and civilian movement into the Ukrainian regions now under Moscow's control.
(Reporting by Reuters; Editing by Gareth Jones)
