"Not every day do we have the opportunity to vote to heal the sick," Democrat Senator Claire McCaskill said, describing it as "a noble cause".
Speaking before the vote, however, a White House spokeswoman said the president would veto the proposed bill because it "crosses a moral line that would use taxpayer dollars to destroy human embryos".
The process of extracting stem cells from an embryo destroys that entity; and conservative Republicans have likened the destruction of frozen human embryos to abortion.
Mr Bush imposed restrictions on spending government money on stem-cell research when he came to power in 2001.
He limited the offer of federal funds to research on stem-cell batches that were already available that August and ruled out funding work on fresh batches.
The president has instead offered to fund an alternative form of research which uses cells taken from amniotic fluid, placentas and from embryos that have died naturally.
The Senate passed a bill endorsing funding for such research later on Wednesday.