Clinton's remarks came nearly three weeks after Brown, an unarmed black teenager, was shot dead by a white police officer. She adopted a careful tone and avoided singling out any one person or entity for blame in an incident that has sparked a national debate over issues of race and the use of force by authorities.
"We can do better. We can work to rebuild the bonds of trust from the ground up," Clinton said.
The potential 2016 presidential candidate applauded President Obama for sending Attorney General Eric Holder to Ferguson, a decision she called "both appropriate and necessary."
Clinton made her remarks at a summit in San Francisco hosted by the data storage company Nexenta. Her comments on events in Ferguson came well after most other prominent politicians weighed in on the shooting and its aftermath.
Brown was fatally shot by Ferguson police officer Darren Wilson on Aug 9. Authorities are still probing details of the shooting, which sparked intense clashes between protesters and police that received widespread attention.
"This is what happens when the bonds of trust and respect that hold any community together fray," Clinton said. "Nobody wants to see our streets look like a war zone. Not in America. We are better than that."
Clinton said her "heart just broke" for Brown's family. "Losing a child is every parent's greatest fear and an unimaginable loss," she said, adding that she also grieves for the city of Ferguson.
Clinton encouraged Americans not to ignore "inequities that persist in our justice system."
"Imagine what we would feel and what we would do if white drivers were three times as likely to be searched by police during a traffic stop as black drivers, instead of the other way around," she said.
The former secretary of state encouraged Americans to join each other in the process of healing after the Ferguson shooting.
"We should all add our voices to those who have come together in recent days to work for peace, justice and reconciliation in Ferguson and beyond," said Clinton.
Martin Luther King Jr.'s "I Have a Dream" speech, Clinton said, should resonate with people and spur them to renew his mission.
"It was 51 years ago today that Dr. Martin Luther King Jr. called us to live out the true meaning of our creed, to make the dream real for all Americans," she said. "And that mission is as fiercely urgent today as when he stood on the steps of the Lincoln Memorial in the hot August sun all those years ago."
This post has been updated