The stampede broke out as thousands of devotees thronged narrow lanes leading to a temple in Shirgao village on Saturday, the Press Trust of India news agency reported.
Tens of thousands of devotees from Goa and neighbouring states of Maharashtra and Karnataka attended the annual Hindu festival at Sree Lairai Devi temple.
The stampede was caused as people standing on a slope near the temple fell over, pushing more people to fall onto each other, director-general of Police Alok Kumar said, according to the news agency.
The injured were taken to nearby hospitals for treatment.Â
Authorities ordered a probe into the incident.
Prime Minister Narendra Modi said he was saddened by the loss of lives and expressed condolences to the victims' families.
"The local administration is assisting those affected," Modi said on social media.
Deadly stampedes are relatively common around Indian religious festivals, where large crowds often gather in small areas.
In January, at least 30 people died and many more were injured in a stampede as tens of thousands of Hindus rushed to bathe in a sacred river at India's massive Maha Kumbh festival, the world's largest religious gathering.
In July, at least 116 people died, most of them women and children, when thousands at a religious gathering in northern India stampeded at a tent camp in Hathras town.
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