Obama, launching a nine-day Asian tour designed to drum up jobs for America, paid symbolic homage to the victims of the Mumbai attack, likening it to the September 11, 2001 strike on the United States.
"We visit here to send a very clear message that in our determination to give our people a future of security and prosperity, the United States and India stand united," Obama said at the Taj Mahal Palace hotel.
"Ever since those horrific days, two years ago, the Taj has been a symbol of the strength and the resilience of the Indian people," Obama said, after touring the rebuilt hotel and meeting survivors of the attacks.
"We will never forget the awful images of 26/11. The flames from this hotel that lit up the night sky. We'll never forget how the world, including the American people, watched and grieved with all of India," said a sombre Obama.
The president said "terrorists" who launched the carnage at the hotel and in the streets of Mumbai offered only death and destruction and could not trump the diversity, tolerance and resilience of nations like India and America.
"We shall never allow that torch of freedom to be blown out, however high the wind or stormy the tempest," Obama said, quoting India's first prime minister, Jawaharlal Nehru.
Treading a fine diplomatic line, Obama did not mention that extremists blamed for the attacks, including the Lashkar-i-Taiba (LeT) group, were based in Pakistan, India's arch-rival and America's anti-terror ally.
The only surviving gunman from the assault said they were recruited, trained and equipped by the banned LeT, with support from elements in Pakistan's military and intelligence service.
On the eve of Obama's departure for India, the US Treasury Department imposed sanctions on the LET and another group, Jaish-e Mohammed (JEM), a key planner of the attacks which killed 166 people, including 31 at the Taj hotel.
Obama and his wife, First Lady Michelle, flew into Mumbai on Air Force One at the start of a trip that will also take the president to Indonesia, South Korea for the G20 summit and Japan for the APEC summit.
The journey, just days after Obama's Democrats took a drubbing in mid-term elections focusing on the economy, is designed to pry open new markets for US exports and create new jobs to ease 9.6 percent unemployment at home.
Security was extremely tight in Mumbai, with roads closed, a ban enforced on sea traffic off the coast and the plaza near the Taj and Gateway of India monument sealed off.
US officials now say that cooperation with India on counter-terrorism is at its highest-ever level, despite India's misgivings about US support for Pakistan.
In Mumbai, Obama will also visit the house where the father of the Indian independence movement, Mahatma Gandhi, stayed on visits to the city. The US president has cited Gandhi as a key influence.
He will also speak to Indian entrepreneurs importing US technologies and top US executives from firms like PepsiCo and GE, before giving a speech to an audience including 200 US business leaders.
A number of major deals are expected to be announced, with Obama lobbying for a series of multi-billion-dollar Indian weapons contracts.
"President Obama intends this trip to be and intends our policy to be a full embrace of India's rise," the president's new national security adviser Tom Donilon told reporters on Air Force One.
Relations between the US and India have warmed considerably in the past 10 years, but there are niggling disagreements over issues such as US controls on the export of "dual-use" technology with military and civil applications.
"The key to the visit from India Inc's point of view is technology, technology, technology," said Amit Mitra, secretary general of the Federation of Indian Chambers of Commerce and Industry.
"The key from the president's point of view is jobs, jobs, jobs -- and the two must meet together during Obama's visit," Mitra said.
Obama moves on to New Delhi on Sunday, where he will meet Prime Minister Manmohan Singh and then deliver an address to parliament.
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