WASHINGTON — The Senate on Thursday approved the most significant overhaul of the nation’s immigration laws in a generation with broad support generated by a sense among leading Republicans that the party needed to join with Democrats to remove a wedge between Republicans and Hispanic voters.

The strong 68-to-32 vote in the often polarized Senate tossed the issue into the House, where the Republican leadership has said that it will not take up the Senate measure and is instead focused on much narrower legislation that would not provide a path to citizenship for the 11 million unauthorized immigrants in the country. Party leaders hope that the Senate action will put pressure on the House.

Leading up to the final votes, which the senators cast at their desks to mark the import of the moment, members of the bipartisan “Gang of Eight,” who drafted the framework of the legislation, took to the Senate floor to make a final argument for the measure. Among them was Senator Marco Rubio, Republican of Florida, who is one of his party’s leading Hispanic voices. When Mr. Rubio finished, the other senators in the group surrounded him on the floor, patting him on the back and offering words of encouragement. “Good job,” one said. “I’m proud of you,” another offered.

The future will show whether voters in Republican presidential primaries share that pride.

After Mitt Romney’s loss in November, top Republicans immediately began formulating a way to improve the party’s standing with Hispanics, who have flocked to Democrats. A group of top Republican political and business officials who support an immigration overhaul met at the downtown Washington office of the anti-tax leader Grover Norquist on Jan. 17 with memories of Mr. Romney’s poor showing in their minds.