Al Jazeera's Barnaby Phillips in New Delhi said: "Congress appears to have done spectacularly well, particularly against the left and forces of the [rival] Third Front, whose supports appears to have collapsed in some areas.
"In the centre of New Delhi, outside the Congress party headquarters, supporters are out on the streets celebrating."
Sonia Gandhi, the Congress leader, said the people had made the right choice.
"First of all I would like to thank the people for reposing faith in the Congress party once again," she said during a joint news conference with Singh.
"The people of India know what is good for them and they always make the right choice."
The UPA is, however, still short of the 272 seats needed for a parliamentary majority.
The BJP-led alliance is projected to take 164 seats while the Third Front, an alliance of communist and regional parties, is set to win only 79 seats, local media said.
Arun Jaitley, leader of the BJP, conceded that his party had been defeated.
"The Congress has the biggest mandate ... I think if Congress wants to compliment some of its leaders in this hour of victory, I think they're entitled to do so," he said.
Prakash Karat, general secretary of the Communist Party of India (Marxist), also conceded defeat.
"We have suffered a major setback," Karat said. "This is a victory for the Congress and its allies who will now clearly form the government."
Focus on rural poor
Sankarshan Thakur, an editor with the Kolkata-based Telegraph newspaper, said the Congress had been rewarded for pursuing policies that focused on the rural poor.
Rajnath Singh, BJP president
Kirti Azad, BJP candidate and former cricketer
Shashi Tharoor, Congress candidate. Former UN undersecretary-general
Shatrughan Sinha, BJP candidate. Bollywood star
Varun Gandhi: BJP candidate. Scion of Gandhi dynasty
L K Advani, BJP party leader
Jayaprada, Samajvadi party candidate. Former film actress
P Chidambaram, interior minister
Manishankar Aiyar, minister of local government
Sajjad Lone, Kashmiri separatist leader who defied boycott call and contested poll
Ram Vilas Paswan, steel minister
It was a combination of luck [good monsoons that helped good crops yield] and good governance. Schemes guaranteeing employment for the rural poor helped," he told Al Jazeera.
Communists in the southern state of Kerala and in the eastern province of West Bengal have put in a poor performance, handing the advantage to the Congress camp, our correspondent said.
"There have [also] been big gains for Congress in Uttar Pradesh, the most populous state in all of India, and Rajasthan, in the west [of the country]," he said.
Raj Chengappa, editor of India Today news magazine, said a number of factors had played a role in the success of the Congress-led combine.
"First I think [was] their entire focus on development and the economy. Congress has a very strong economic team. Second, they have shunned divisive politics, unlike the BJP," he told Al Jazeera.
"Third, we had for the first time a very young generation that is voting in large numbers. I went to a poll booth and I was surprised that almost 50 per cent of the voters were below 25. This is an unusual phenomenon. I think they rooted for Congress.
"Another factor was that rural India voted strongly for development again, because the [ruling coalition] had money in their hands; the economic meltdown had not hit [rural dwellers] as much as [those] in urban India."
Computerised counting of votes at hundreds of centres across the country began at 0230 GMT on Saturday, with the results for all parliamentary seats due later in the day.
About 714 million people were eligible to vote in the largest such exercise in the world.
The vote was staggered over a month to allow security forces and election officials to supervise it.