The National Disaster Management Centre said two people were buried alive in a mudslide at Walapane while another was killed in Meepai.
Elsewhere local relief workers in Hanguranketha and Nuwara Eliya areas said another 11 were killed when unstable mountain slopes crashed on several homes.
A total of 61,281 people were driven out of their flooded homes, the disaster management centre said.
Flooding in the south was compounded by irrigation reservoirs which overflowed after the heavy rains and helicopters were deployed to evacuate marooned residents, officials said.
Sri Lanka depends on monsoon rains for farming as well as power generation.
However, the seasonal downpours frequently also cause loss of life and damage to property in low lying areas.
The two main monsoons in Sri Lanka are from May to September and December to February.