Two people were stabbed to death and one was killed by police gunfire in the old part of the city, oficials said.
The trouble erupted on the 11th anniversary of the razing of the Babri mosque in Ayodhya, northern India.
Security was tightened across India in preparation for the anniversary of the mosque's demolition by Hindu activists.
The violence on Saturday night followed a relatively peaceful day of protests across the country by Muslims, and celebrations by Hindu groups.
Groups of Hindus and Muslims reportedly started throwing stones and attacking each others' houses.
"We had to open fire to control the situation as the mobs were armed with petrol bombs, swords and iron rods," said the city's police commissioner RP Singh.
"We are monitoring the situation carefully. Additional troops have been rushed into all the trouble spots and we have imposed a curfew in these areas."
Deployed troops
There were no reports of violence anywhere else in the country following heightened security ahead of the anniversary.
The razing of the Babri mosque in 1992 triggered a year of violence that killed 2,000 people across India.
Thousands have been killed in Hindu-Muslim riots across the country since then.
Hindu activists claim the mosque was built on the site of an earlier temple to Lord Ram, but Muslims dispute this.
Legal arguments are continuing in an attempt to determine ownership.