The burst happened in Solai near the Rift Valley city of Nakuru, 120 miles northwest of the capital Nairobi, late on Wednesday.
The Kenyan Red Cross estimates that more than 500 families have been caught up in the deluge, with many people still unaccounted for.
Police chief Gideon Kibunjah said 36 people had been hospitalised.
Local reports suggest roughly 100 homes have been affected, including properties on the Nyakinyua Farm that borders the reservoir.
Witnesses at a nearby shopping centre said they heard a huge explosion before water gushed out of the Patel dam, the Kenyan news site Standard Digital reports.
Nakuru governor Lee Kinyanjui said: “We will do our best to take affected families to safety and help them get medical attention.”
The disaster comes after weeks of heavy rain.
It has reportedly caused billions of Kenyan shilling’s worth of damage, with the force of the water levelling nearby structures.
A billion Kenyan shillings is worth roughly £7.3million.
Mr Kinyajui said: “The water has caused huge destruction of both life and property.
“The extent of the damage has yet to be ascertained.”
More from Kenya
Surgeons open ‘wrong man’s head’ during operation in Kenya
Kenyan newspaper sorry for fake death notice of high-profile political donor
Leading ivory and rhino horn investigator found fatally stabbed in Kenya
Kenya’s opposition leader Raila Odinga ‘sworn in’ as TV stations taken off air
Kenya blames colonial Britain for child landmine deaths
Ultra-runners to risk lives to save Kenyan rhinos
Kenya Red Cross said on Twitter it had so far rescued 39 people.
Recent flooding in the country has already killed more than 152 people and displaced a quarter of a million others.