The deaths occurred as a sudden downpour sent people waiting to collect food at a government-run school scurrying for cover, police said.
The incident at the relief camp for flood victims also left another 50 people injured, some of them seriously, a police spokesman said.
This was the second incident of its kind in Chennai, formerly known as Madras, in the past two months.
Six people were killed and 20 injured last month in a stampede near a relief camp in a northern district of Chennai.
At least 430 people have been killed in Tamil Nadu state and 500,000 left homeless by storms that have pounded the state since late October.
The state receives most of its annual rainfall from the northeast monsoon, which is often marked by cyclonic activity in the Bay of Bengal.