By voice vote, the Senate adopted the legislation sponsored by Senate Foreign Relations Committee Chairman Bob Menendez and pushed by the panel after the Feb. 12 test.
The bill urges the Obama administration to work with the United Nations on tougher sanctions against North Korea. The administration is currently negotiating in the U.N. Security Council for tougher penalties against Pyongyang.
The underground atomic blast earlier this month was the third conducted by North Korea since 2006. It was widely condemned by the United States and the international community.
Menendez has said North Korea's program is a threat to U.S. national security and regional peace and stability.