She said that she hopes to find areas of common interest with Russia, while working to resolve differences between Washington and Moscow.
"The United States is very much hoping that we can find areas of agreement and co-operation. Will there be disagreements? Of course there will be," she said.
"But on so many important matters that confront us, we think that the United States and the EU and Russia can be partners."
Clinton spoke during a debate in Brussels attended by Hans Gert Poettering, president of the European Parliament, and young political activists.
She also spoke about crises facing the world, and the Middle East peace process, among other issues.
Poettering said Clinton offered the chance to "rebuild American foreign policy and to restore your country's influence".