The Senate easily passed the most sweeping changes to immigration law in nearly 30 years, sending the landmark measure to the House, where conservative lawmakers threaten to slow the drive to grant legal status to many of the estimated 11 million people living illegally in the U.S.

The 68-32 vote Thursday marked a major step in a long-debated overhaul to the immigration system and drew ceremonial flourish. Vice President Joe Biden presided over the proceedings, and lawmakers rose from their desks to cast their votes, a rarely...