Mr Vajpayee, the BJP's tallest leader, was admitted to hospital on June 11 with a kidney tract infection and chest congestion.
But in India he was hailed as a hero, and it helped him bolster his image as a leader who was serious about national security and not afraid of worldwide pressure.
Mr. Karzai sat through the funeral alongside Sri Lankan Foreign Minister Lakshman Kiriella and met a cross section of dignitaries in attendance.
From Moscow, Russian president Vladimir Putin in a message to his Indian counterpart Ram Nath Kovind said: "Atal Bihari Vajpayee rightly commanded great respect around the world".
Indian politicians and celebrities, including cricket captain Virat Kohli and opposition Congress party leader Rahul Gandhi, took to Twitter to post their condolences.
"I am deeply saddened by the passing away of former prime minister and Bharat Ratna Atal Bihari Vajpayee".
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"France offers its condolences to his family as well as the Indian authorities and people", it said.
"Today, our two countries and our bilateral relationship continue to benefit from Prime Minister Vajpayee's vision, which helped promote expanded cooperation", Pompeo said.
However, he also defended a Hindu campaign for construction of a temple on the disputed site, which Hindus believe was the birthplace of god-king Rama. He had suffered a stroke in 2009, which reportedly weakened his cognitive abilities, following which he developed dementia. Leaders of the BJP were gathering at his home in New Delhi to pay their respects.
A government spokesperson said Chief Minister Yogi Adityanath has ordered immersion of Vajpayee's ashes in every district as Uttar Pradesh was his "karma bhumi" and every region of the state had a close link with him.
During his second stint as prime minister in 1998, he surprised the world by announcing that India had conducted a series of underground nuclear tests.
"As Foreign Minister, Mr Vajpayee strived to open up ways towards normalisation of relations among neighbouring states and had successfully taken this agenda to the new heights after assuming Office of the Prime Minister", he added.