The footage shows one of the students beaming as he lifts the dog by the scruff of the neck to the ledge and finally tosses her over. The other student is thought to have filmed the incident.
An animal rights activist later found the dog injured but alive.
Police in Tamil Nadu state registered a case against the two men after an activist from the government’s Animal Welfare Board of India filed a complaint on Tuesday.
But police said the pair, who have been identified as final-year students of a medical college in the state capital Chennai, had gone missing.
“The two medicos were missing from their house. A team has been sent to their native place to trace them,” said local police inspector, Frank D. Ruben.
Activist Shravan Krishnan posted a photo on Instagram late Tuesday by of him holding the injured animal.
“Good news – The dog has been found alive. She is not able to walk properly. She is with us now. Will be under treatment. What breaks my heart is that she was still wagging her tail,” he wrote in the caption.
Earlier, the Indian branch of the Humane Society International offered a reward of 100,000 rupees ($1,480) to anyone who could provide information for the students’ arrest.
Others called for stronger animal cruelty laws in the country.
In 2007, animal rights supporters created uproar when municipal workers in the southern Indian high-tech city of Bangalore culled thousands of street dogs after a child was mauled to death by a pack of neighbourhood strays.