CHICAGO — The fierce struggle among Republicans over whether to make Medicaid available to more low-income people played out in Michigan on Tuesday as the Republican governor, Rick Snyder, narrowly succeeded in swaying enough conservative senators in the State Legislature to accept the expansion, which was part of President Obama’s health care law.

Mr. Snyder’s preferred bill — one he had lobbied for intensely for months — initially fell short by one vote, but the governor salvaged a deal hours later. The vote in the Republican-controlled Senate was 20 to 18, with only 8 Republicans in favor. The Michigan House, which had earlier approved a similar measure, will need to vote on the Senate version before Mr. Snyder can sign the bill.

“The Affordable Care Act has probably been one of the most divisive issues that our country has faced in the last few years, and many people do have strong opinions both for and against,” Mr. Snyder said after the vote. “I just ask that all Michiganders step back and look to say this isn’t about the Affordable Care Act. This is about one element that we control here in Michigan that we can make a difference in here in people’s lives.”

While the authors of the federal health care law intended to expand Medicaid, the federal and state health program for poor people, and at least initially pay for the expansion, the Supreme Court ruled in 2012 that states could opt out, setting up a struggle that has played out in the states largely along partisan lines.

Like Mr. Snyder, some Republican governors have found themselves at odds with their own party’s legislative caucuses in state capitals like Lansing that are dominated by Republicans.

In Arizona, which eventually approved an expansion, Gov. Jan Brewer found vehement opposition from some lawmakers. In Florida, legislators have resisted expansion, despite Gov. Rick Scott’s support. And in Ohio, Gov. John R. Kasich’s push for expansion has so far not been successful.

For months, the fight in Michigan, which has the nation’s 10-largest uninsured population, has been intense. Mr. Snyder, a former businessman in his first term, said the expansion would ultimately save money, control medical costs and help the state’s economy. That pitted him against more conservative members of his own party, and led some Tea Party leaders in the state to say he will lose support if he seeks re-election next year.

On the floor of the Michigan Senate on Tuesday, the debate was heated, though lawmakers said the discussions — particularly those within the Republican caucus — had been even more tense behind closed doors.

Advocates praised the measure as fiscally sensible for the state, given the promise of federal money, and crucial for hundreds of thousands of low-income residents without insurance.

Already, Medicaid covers more than 1.8 million people in the state, Michigan officials said, and the expansion would ultimately grant coverage to more than 400,000 others. People making up to 133 percent of the federal poverty level — or about $15,500 a year for a single person — would newly be covered.

“It’s a benefit to every person in the state of Michigan,” said State Senator Gretchen Whitmer, the Democratic leader, said on the floor. “It’s good public policy, and it makes good fiscal sense.”

Senator Roger Kahn, a Republican, told his colleagues, “This is not Obamacare or the Affordable Care Act.” Instead, he argued, the measure will reform the costs of medicine across the state and become what he described as “a national model” for other states.

But opponents said a Medicaid expansion would represent tacit approval of Mr. Obama’s health care law. They said it would encourage big government and be an irresponsible promise of spending by Michigan in the years ahead. Senator Mike Green, a Republican, described the plan as a promise of “federal funny money.”

And Senator Patrick J. Colbeck, another Republican opponent, said, “We’re spending money we do not have,” adding, “And we’re forcing decisions right now onto our youth.”

In June, the Michigan House approved the Medicaid expansion with support from Democrats and enough members from that chamber’s Republican majority. But the State Senate, where Republicans hold 26 of 38 seats, moved more slowly, with some conservative Republicans openly rejecting Mr. Snyder’s views, refusing to call a vote and proposing alternatives.

At one point, Mr. Snyder, who regularly promotes a gentle-sounding political philosophy of “relentless positive action,” flew home early from a trade mission to Israel and had uncharacteristically sharp words for the Senate, telling them to “take a vote, not a vacation.”

In recent days, the struggle intensified. The governor’s office took part in urgent, private conversations, lawmakers said, while advocates for and against the measure led demonstrations, ad campaigns and phone banks around Michigan.

“We firmly believe that a vote to support Medicaid expansion is a vote to support the president’s health care law,” said one opponent, Annie Patnaude, deputy state director of Americans for Prosperity-Michigan. Protesters on the other side of the issue appeared in Lansing on Tuesday, saying it was high time for a vote.