Followers of an upstart Indian political party danced in the streets on Tuesday after inflicting a crushing defeat on Prime Minister Narendra Modi's right-wing party in a key election in the capital.

The poll was the first electoral test for Modi's Bharatiya Janata Party (BJP) after it passed a controversial nationality law which opponents say is anti-Muslim.

Aam Aadmi Party (AAP) supporters cavorted to bhangra music and set off fireworks as the vote count showed they had crossed the 36 seats needed to secure a majority in the 70-seat regional assembly.

With more than 75 per cent of the votes counted, the AAP was leading in 63 of the 70 constituencies and the BJP in seven, the Election Commission said.

The BJP was projected to win 13 seats, up from three in 2015 but far below its own expectations. The party's local chief Manoj Tiwari had predicted it would win a majority.

Hindu-nationalist Modi, whose party swept to power in national elections last year, congratulated AAP leader Arvind Kejriwal, the incumbent Delhi chief minister.

“Wishing them the very best in fulfilling the aspirations of the people of Delhi,” Modi tweeted.

The BJP had launched an aggressive campaign to win the city of nearly 20 million people from the AAP, using the election to rally support for the law easing citizenship rules for religious minorities from Bangladesh, Pakistan and Afghanistan, but excluding Muslims.

At least 25 people have been killed in protests over the legislation so far.

Explainer: What does India's new citizenship law mean?

But the AAP, which swept to power in 2015 after it was launched three years earlier by former tax officer Kejriwal, retained control in an impressive showing.

The defeat in Delhi is the latest in a string of setbacks for the BJP at regional elections over the past two years.

Kejriwal, 51, fought the election on local issues such as subsidised water and electricity, as well as the safety of women.

“This win has given birth to a new type of politics — the politics of work,” he told cheering supporters at party headquarters.

“This is the type of politics that will take the country forward in the 21st century.”

Yogendra Yadav, an academic who was a member of the AAP executive until 2015 and now has his own party, said the result was a clear rejection of Modi and his party's angry campaign.

“The BJP indulged in one of the most vitriolic, communal hate-mongering campaigns as a desperate electoral gamble,” he told AFP.

“If this succeeded, it would have become a template for everyone else to follow.”

Congress, led by the storied Gandhi-Nehru dynasty and the main opposition at the national level, was set to draw a blank in another low for a party that ruled Delhi for 15 years before AAP took over.

Final results from the Election Commission of India were not expected until late Tuesday.

Setback

A defeat is a setback to Modi’s prestige, coming less than eight months after he led the BJP to a resounding victory in national elections. The party won all seven of the capital’s parliamentary seats in those polls.

The victory will be a major boost for Kejriwal, an anti-corruption crusader.

Neelanjan Sircar, an assistant professor at Ashoka University near New Delhi, said that local issues, including delivery of basic services like education and health, appeared to sway voters towards the AAP, even as the BJP ran a polarising campaign on the back of Modi's image.

“Modi is a larger than life character at the national level, which obviously gives the BJP a huge advantage in national politics,” Sircar said.

“But it doesn't translate to state-level politics, where the BJP often doesn't have a charismatic face.”