Some Republicans had been critical of the legislation because of several provisions, including one to prohibit discrimination of victims of domestic violence because of their sexual orientation. Another controversial part of the bill would broaden the authority of Native American tribes to handle cases of domestic violence.
After the votes the Senate planned to move immediately to a final vote on the bill.
The progress to final passage of the bill comes a day after Senate Republicans agreed on Wednesday to allow the legislation to come to a final vote and pass.
—This story was updated at 4:31 p.m.