Although Russia's crisis-hit economy is barely one-eighth of the size of America's, Mr Putin spoke the language of parity between the two nations.
The comments came in separate addresses to foreign leaders and to the Russian public, following by far the most difficult year for Moscow's relationship with the West since the Cold War.
In a generally conciliatory message, Mr Putin used the coming 70th anniversary of the end of the Second World War to invoke the memory of Russia fighting "shoulder to shoulder" with the Western Allies against Nazi Germany.
"This historic date is a reminder of the responsibility of Russia and the United States in maintaining peace and international stability," he said in a message to President Barack Obama. Mr Putin added that he wanted relations with America to "develop successfully proceeding from principles of equality and mutual respect".
Mr Putin avoided direct references to the confrontation with the West over Ukraine, opting instead for diplomatic language emphasising mutual interests.
In his note to David Cameron and the Queen, Mr Putin said he was counting on "constructive cooperation" between Russia and Britain that "will enable us to guarantee peace and security in Europe and the whole planet."
But one world leader conspicuously absent from the list of messages was President Petro Poroshenko of Ukraine - although Mr Putin made a point of congratulating the leaders of every other post-Soviet state.
In his address to the Russian public, Mr Putin did not skirt the subject of Ukraine. He praised Crimea's "return home" after Russian troops annexed the peninsula from Ukraine in March.
"Love of one's country is one of the most powerful and uplifting feelings," said Mr Putin. "It manifested itself in full in the brotherly support to the people of Crimea and Sevastopol, when they resolutely decided to return home."
In the televised address, which is traditionally broadcast just before midnight in each of Russia's 11 time zones, Mr Putin urged his compatriots to be prepared to work hard in the coming year - in an apparent acknowledgement of the growing economic difficulties.
"This year will be exactly what we make of it - there is no other way," he said.
The messages came a day after Mr Putin's government jailed the brother of a prominent opposition activist.Oleg Navalny, the brother of the anti-corruption campaigner and Kremlin critic Alexey Navalny, was imprisoned for three and a half years for fraud on Tuesday.
Alexey Navalny himself was handed a lenient suspended sentence for the same period.
Police arrested over 100 people in central Moscow on Tuesday night after about 2,000 protesters angered by the sentence gathered near the Kremlin.
Alexey Navalny himself broke house arrest to try to attend the protest, but was arrested before he could join the main demonstration.
Several dozen more protesters, including members of the punk group Pussy Riot, were arrested on Wednesday morning after spending the night in a giant Christmas tree bauble set up on Manege Square.
