The 63-37 vote left it unlikely the Congress would be able to gather the requisite two-thirds vote in both the Senate and House of Representatives to override a veto, which would be the president's first in five and a half years in office.
Mr Bush has said he would veto the bill on the grounds that it giving public finance for research that involves the destruction of human embryos.
"The president believes strongly that for the purpose of research it's inappropriate for the federal government to finance something that many people consider murder. He's one of them," Mr Bush’s spokesman Tony Snow said.
The controversial Stem Cell Research Enhancement Act would lift rules set by Bush in 2001 that make federal funds available only to research on a small number of embryonic stem-cell lines which existed at that time.
Government money was barred from supporting work on any new lines derived from human embryos - a restriction that opponents say hampers overall research.
They say that some of the stem-cell lines approved under the 2001
legislation are unusable.
Stem-cell research advocates say the technique shows promise for the treatment of degenerative diseases such as Alzheimer's and Parkinson's, and for diabetes.
But opponents consider the destruction of human embryos morally wrong.
According to recent public opinion polls, 70 percent of Americans support embryonic stem-cell research.
The Senate bill earned the support of Nancy Reagan, the widow of late Republican president Ronald Reagan who suffered from Alzheimer's disease, and actor Michael J. Fox, afflicted with Parkinson's.
Mr Bush's position is backed by a strong and vocal minority of social conservatives, including Christian groups.