The animals entered a suburb in Mysore, in the southern Karnataka state, from a nearby forest and soon caused widespread panic and havoc.
A 55-year-old man was killed after coming out of his house to see what the commotion was all about.
He was crushed against the ground by one of the animals and died instantly.
And on another street two cows were tossed around and eventually killed by the huge beasts in front of a crowd of onlookers.
Karnataka state higher education minister SA Ramdas said schools and colleges were closed for the day and extra police deployed as a precaution.
One elephant barged into a women's college compound and roamed menacingly in the grounds, while the other got into a residential area.
After three hours, forest rangers and staff from Mysore Zoo managed to capture and tranquilise the pair.
One official blamed the rampage on the encroachment of human settlements into forested areas that are the elephants' natural habitat.
"Unregulated expansion of farm lands and increasing movement of people and transport vehicle through the elephant corridor are making the wild jumbos enter into villages and towns in search of food and shelter," he said.
The two captured elephants will be released back into the wild later on Wednesday, Mr Ramdas said.
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