Singh, a 71-year-old Sikh who will be India's first non-Hindu prime minister, immediately pledged to turn the world's largest democracy into an economic model that "makes new opportunities available to the poor."
President Abd al-Kalam, the ceremonial head of state, invited Singh on Wednesday to replace the government of Hindu nationalist premier Atal Behari Vajpayee after a tumultuous day following Gandhi's decision not to be prime minister.
Gandhi, the Congress party president who engineered the upset victory by targetting Indians left out of the country's surging economic growth, said "the government will be safe in the hands of Mr Singh."
Italian-born Gandhi had faced a virulent campaign by Hindu hardliners against her foreign roots and family pressure not to become prime minister.
Her husband Rajiv Gandhi and mother-in-law Indira, who had both served as premier, were assassinated.
Singh, an Oxford-trained economist and former official of the International Monetary Fund, ended decades of protectionism as finance minister in the last Congress government from 1991 to 1996.
Singh (L) stepped forward when
Sonia Gandhi (R) declined the post
"We have always said that economic reforms with emphasis on human elements will continue," Singh said.
"We will give to the world and to our people a model of economic reforms which adds to the process of development to bring new opportunities for the poor and downtrodden," Singh said, standing alongside Gandhi at the British-built presidential palace in central New Delhi.
The Bombay Stock Exchange, which had crashed after Gandhi and Singh's communist-backed coalition defeated the market-friendly government, rallied up 2.65% on expectations the economy would be in safe hands under Singh.
"The father of India's reform programme rising to the prime ministership would be very positive from the standpoint of the market," said PK Basu, head of Robust Economic Analysis.
"But I would caution against excessive euphoria since Doctor Singh as an economic reformer is well regarded, but his abilities as a political manager are untested."
With backing from leftists and regional parties, Singh will enjoy a solid majority of 325 seats among the 539 members of parliament elected so far, according to a Press Trust of India tally.
The Communist Party of India, which has pledged support to a Congress-led government coalition, said the overriding concern was to stop Hindu nationalists from returning to power and it would support Singh.
"He is one of the most decent persons, a knowledgeable economist and I will opt for him any time over any person in the Bharatiya Janata Party," said communist Somnath Chatterjee.