The village of Kurichedu has been placed in lockdown after a local surge in cases of coronavirus.
District police chief Siddharth Kaushal said the people who died had mixed the sanitiser with water and soft drinks.
They were alcohol-dependent, he added, and had started drinking the mixture about 10 days before they died.
"We are investigating whether the sanitiser had any other toxic content," Mr Kaushal told reporters.
He added that they had sent samples of the sanitiser away for chemical analysis.
"Some people who are heavily addicted to alcohol had been consuming hand sanitisers for the high," he told Reuters news agency.
"Alcohol is not available because of the lockdown, but hand sanitisers are easily available."