The woman is heard screaming for mercy as the police beat up the man.
The incident, believed to have taken place on Monday night at the Andheri police station, has sparked anger.
"Let me go," the woman pleads repeatedly, but the blows don't stop.
The two are reportedly colleagues at a call centre.
The police allege that they were found drunk and fighting on the road. When they were taken into the police station nearby, the police claim, they began hugging.
"We saw that they were drunk and hugging each other and that too outside the police station. We felt that it would turn into a case of outrage of modesty and we tried to separate the couple. Police did not beat them up and they haven't filed a complaint," said Dilip Rupwate, Assistant Commissioner of Police, Mumbai Police.
A cop at the police station said that the video was recorded to damage the reputation of the police.
The couple was allegedly kept in the police station through the night and allowed to go in the morning with their parents. The Maharashtra Government has now lodged an inquiry into the incident.
The Mumbai Police has been battling criticism of excesses after two more incidents recently.
Last month, two Muslim teenagers were assaulted in Bandra in west Mumbai by policemen who accused them of being Pakistan terrorists.
Before this, a young woman was beaten for trying to use a gate meant for VIPs to enter a Ganesh festival venue.
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