Officers named the victim as Yusra, aged 17, saying she was shot twice by the man who broke into her home and also wounded her mother late on Wednesday in the Nizamuddin residential and commercial district.
"During the day, Yusra objected to him urinating at the gate of the building in which both lived," additional police commissioner Ajay Chaudhry said.
Chaudhry said the 21-year-old man then returned with a handgun the same evening and shot at Yusra and her mother in a bedroom of their working-class home in Nizamuddin, a Muslim-majority enclave.
The senior police officer described the gunman as a "jobless vagabond".
"He is absconding but he is somewhere out there and we will get him," he added.
The Indian capital saw a spate of shootings in September. In one incident, a 23-year-old shot dead his ex-girlfriend, her landlady and then drove to a city suburb to kill her father and sister before turning the gun on himself.
A few days later, two friends went on a shooting rampage which left three women and two young girls dead.
Figures for gun seizures in New Delhi show a rising trend.
Police in 2009 seized 573 illegal weapons, in 2010 there were 634 and in 2011 the number touched 770. In the first nine months of this year, officers have already recovered 594 guns.