With large parts of central Delhi sealed off after a wave of violent protests, Mr Singh said he understood the anger over the "monstrous crime", but called for demonstrations to remain peaceful.
"There is genuine and justified anger and anguish at this ghastly incident," he said in a televised address.
"I feel deeply sad at the turn of events leading to clashes between protesters and police forces.
"I assure you that we will make all possible efforts to ensure security and safety to all women in this country.
"As a father of three daughters myself, I feel as strongly as you. We will ensure justice is delivered."
His pleas for calm were echoed by the victim's father.
"Please pray for my daughter and do not not resort to any violent activity. She needs your support and your prayers," the father, who cannot be named for legal reasons, told the Hindi-language News 24 network.
"My daughter is fighting for her life, but she is very strong and has a strong will to live."
Protests have been taking place across India over the past week to denounce the police and government over the gang-rape of the 23-year-old student in the capital and the surge of violence against women.
Official figures show that 228,650 of the total 256,329 violent crimes recorded last year were against women.
Delhi has been dubbed the rape capital of India with the number of rapes in the city rising 17 per cent to 661 this year.
The biggest protests were seen in New Delhi on Saturday, prompting police to cordon off areas around government buildings.
However fresh clashes erupted on Sunday, which the police control room in New Delhi said had left more than 100 people injured, including 60 policemen.
During those protests, police used water cannons to disperse demonstrators.
The clampdown was further tightened on Monday, with commuters having to make lengthy detours to make it to work.
In north-eastern Manipur state, an Indian journalist was killed after police opened fire on a crowd protesting the sex attack.
The victim of the Delhi gang-rape remains in a critical condition in hospital after suffering terrible injuries during her assault on a bus which began when she and a male companion were picked up after a night out at the cinema.
Police and prosecutors say six men, who were drunk and were joy-riding in an off-duty bus with tinted windows, took turns in raping the student at the back of the bus before throwing her off the vehicle.
During her assault, the victim suffered serious intestinal injuries from being beaten with an iron rod.
All six of the alleged attackers have now been arrested and remanded in custody.
Home minister Sushil Kumar Shinde has said the Congress-led government would look at "enhanced punishment in the rarest of rare rape cases" - a reference to capital punishment.
India has executed two people since 2004 - one of whom was Mohammed Kasab, the sole surviving gunman from the 2008 Mumbai attacks. The other was a criminal who raped and killed his schoolgirl victim.