US President Barack Obama also issued a written statement condemning the deadliest attack on India's commercial capital since the traumatic 2008 assault by Islamist militants.
"I strongly condemn the outrageous attacks in Mumbai and my thoughts and prayers are with the wounded and those who have lost loved ones," Obama said.
Clinton said the attacks, which left at least 21 people dead and more than 140 injured, had been "designed to provoke fear and division" and said: "Those who perpetrated them must know they cannot succeed.
"I will be traveling to India next week as planned," she said. "I believe it is more important than ever that we stand with India, deepen our partnership, and reaffirm our commitment to the shared struggle against terrorism."
Three bombs exploded in busy districts of southern Mumbai at around 7:00 pm local time. It was the same area targeted two and half years ago by suspected Pakistani militants who caused mayhem and bloodshed during a 60-hour siege that left 166 people dead.
No group claimed responsibility for the latest attacks, though suspicions initially fell on two Islamist groups that have targeted India in the past: the home-grown Indian Mujahideen and the Pakistan-based Lashkar-e-Taiba (LeT).
"The Indian people have suffered from acts of terrorism before, and we have seen them respond with courage and resilience," Clinton said.
"Our hearts are with the victims and their families, and we have reached out to the Indian government to express our condolence and offer support."
At a joint Washington press conference with Clinton, Foreign Minister Sergei Lavrov of Russia also denounced the attacks, saying: "We condemn the people who organized this act and we extend our condolences to India."
Clinton will next week become the most senior US official to visit India since a state visit by Obama last November.
She will travel to New Delhi to attend the second India-US Strategic Dialogue on July 19 after presiding over the first such meeting between the world's two largest democracies last year in Washington.
Clinton will also visit the business hub of Chennai during her stay but was not due in Mumbai, according to the schedule provided to journalists.
Security concerns are likely to top her meetings, as India expresses growing concern over instability in its neighbor and arch-rival Pakistan, especially in the wake of the latest Mumbai attacks.
US and Indian investigators believe that the 2008 assault was carried out by Pakistan-based Islamic militants. Indian Prime Minister Manmohan Singh ruled out reprisals at the time despite public pressure.
Clinton last visited the South Asian region in late May when she flew into Pakistan and urged Islamabad to take decisive steps to defeat Al-Qaeda after relations between the wary allies went into freefall following the US commando raid that killed Osama bin Laden on May 2.
US lawmakers also condemned the Mumbai bombings. "Senseless attacks such as these shock civilized people everywhere," said Democratic House leader Nancy Pelosi.
"There can be no justification for what took place, and the targeting of civilians on their commute home from work is deplorable," said Representatives Ed Royce and Joe Crowley, co-chairs of the congressional caucus on India.