Both Mukherjee and Manmohan Singh said the victim's death in a Singapore hospital would not go in vain, and that India should become a safer place for women.
The 23-year-old, who suffered multiple organ failure after the Dec 16 rape, passed away at 4.45 a.m. (2.15 IST) with her distraught family and Indian diplomats by her side at Singapore's Mount Elizabeth Hospital.
Condoling her death, President Mukherjee called the victim, a trainee physiotherapist who worked in Delhi, a "true hero".
"I am deeply distressed by the unfortunate demise of the girl who passed away."
Mukherjee said: "She was a brave and courageous girl who fought till the very last minute for her dignity and her life. She is a true hero and symbolizes the best in Indian youth and women.
"I convey my condolences to her parents and family who have faced this adversity with fortitude and grace. I pray that they have the strength to bear this loss.
"The nation will mourn the passing of this brave daughter of India."
Stressing that her passing away should not be in vain, the president said the government would "do everything possible to ensure that such an incident never happens again".
He urged "everyone to maintain peace and calm" and told the "authorities to bring the perpetrators of this ghastly crime to justice".
Recognising the "emotions and energies" the gang-rape had generated in the capital and elsewhere, Manmohan Singh said India genuinely desires change.
"It was up to us all to ensure that her death will not have been in vain", he said as he joined the nation "in conveying to her family and friends" his deepest condolences at this terrible loss.
"I want to tell them and the nation that while she may have lost her battle for life, it is up to us all to ensure that her death will not have been in vain.
"We have already seen the emotions and energies this incident has generated. These are perfectly understandable reactions from a young India and an India that genuinely desires change," he said in a message.
It would be a true homage to her memory, Manmohan Singh added, if people channelise their emotions and energies "into a constructive course of action".
"The need of the hour is a dispassionate debate and inquiry into the critical changes that are required in societal attitudes.
"The government is examining, on priority basis, the penal provisions that exist for such crimes and measures to enhance the safety and security of women," he said in his statement.
The prime minister expressed the hope that "the entire political class and civil society will set aside narrow sectional interests and agenda to help us all reach the end that we all desire - making India a demonstrably better and safer place for women to live in".
"I pray for the peace of the departed soul and hope that her family will have the strength to bear this grievous loss."
Delhi Chief Minister Sheila Dikshit appealed for calm. She described the woman's death as "truly, terribly sad news".
"In being calm and peaceful... we can reflect what we need to do in future," she added.
The horrific gang-rape, in which the rapists used an iron rod to torture her, triggered angry demonstrations across India against growing crimes against women. A policeman died in one such protest in Delhi.