Churkin died yesterday, a day before his 65th birthday.
"Farewell Friend. Russia loses a stalwart diplomat; the world of diplomacy a consummate practitioner; India a friend. RIP Amb. Vitaly Churkin," India's Permanent Representative to the UN Syed Akbaruddin said in a post on Twitter.
Akbaruddin's message was accompanied with his picture with Churkin taken at an event at India's Permanent Mission here.
The Russian Permanent Mission in a statement said Churkin was in his office "fulfilling his duties" when the tragedy occurred. It did not give details about the cause of his death but it is understood Churkin was rushed to hospital after a cardiac arrest.
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UN Secretary-General Antonio Guterres said he "learned with shock and sadness" about the Russian Ambassador's death.
Guterres said Churkin was an outstanding, uniquely-skilled diplomat, a powerful orator with great wit, and a man of many talents and interests.
"Although we served together for a short time, I greatly appreciated the opportunity to work with him and will deeply miss his insights, skills and friendship," he said.
Guterres's predecessor Ban Ki-moon, in a statement from Seoul, said during his 10 years as Secretary-General it was a "privilege" to work with Churkin and watch him represent Russia with passion and dedication.
"His diplomatic skills, quick wit, and ready sense of humour will long be remembered by those who knew him and by those who will study the history of the United Nations in the years to come," Ban said.
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The news came around the time of the daily noon briefing at the UN and deputy spokesperson Farhan Haq was visibly shocked when a journalist mentioned that Churkin has died.
UN General Assembly President Peter Thomson said: "His name will live on in the annals of UN history. We will remember him for his intellect and diplomatic skill, his good humour and consideration for others. We all experienced and respected the pride he took in serving his country and the passion and resolution he brought to his job."
Appointed the Permanent Representative to the UN in 2006, Churkin was one of Russia's longest-serving ambassadors to the UN.
In December, when the Council argued on actions in Syria, former US envoy Samantha Power asked Syria and its allies Russia and Iran whether they are "truly incapable of shame?"
Retorting Power's remarks, Churkin said she was speaking "as if she was Mother Teresa".
Yesterday, Power tweeted, "Devastated by passing of Russian UN Amb Vitaly Churkin. Diplomatic maestro &deeply caring man who did all he cld to bridge US-RUS differences".