Mitt Romney and Rick Santorum each won Republican presidential primaries in multiple states on Tuesday night, with Romney narrowly edging his rival in the key state of Ohio after a battle that highlighted stubborn divisions in their party.

Former House speaker Newt Gingrich won the primary in his home state of Georgia, once again reviving his campaign. Rep. Ron Paul of Texas did surprisingly well in a losing effort in Virginia, indicating that the tumultuous four-way GOP race is likely to rumble on for weeks.

Romney beat Santorum by just one percentage point in Ohio, a state that is vital to Republican hopes in November’s general election. Romney had trailed badly there in recent weeks, but rebounded as a result of heavy TV advertising and repeated visits to the state. He also won four states where he faced little opposition: Massachusetts, Virginia, Vermont and Idaho. In the Alaska caucuses, he won with 32.6 percent of the vote, compared to 29 percent for Santorum, 24 percent for Paul and 14.2 percent for Gingrich.

Each victory helped Romney add to his lead in delegates, the tally that will ultimately determine the GOP’s nominee. But the former Massachusetts governor, who has struggled to capture the passion of Republican voters, acknowledged that it could be a struggle for him to clinch the nomination before the Republicans’ nominating convention.

“We’ve got the time and the resources and a plan to get all the delegates and we think that’ll get done before the convention,” Romney said on CNBC’s “Squawk Box “ on Wednesday. “But one thing I can tell you for sure: There’s not going to be a brokered convention, where some new person comes in and becomes the nominee. It’s going to be one of the four people that are still running.”

In his victory speech Tuesday night, Romney focused on President Obama, paying little heed to his GOP rivals. “We’re counting up the delegates, and that looks good,” Romney said. “And we’re counting down the days until November, and that looks even better.”

But Santorum scored wins in Oklahoma, Tennessee and North Dakota--victories which, along with his close loss in Ohio, underscored his ability to draw conservative and blue-collar voters with a bargain-basement campaign.

“The Republican Party has to nominate somebody who can talk about the broad vision of what America is,” Santorum told supporters in the small city of Steubenville, Ohio. To beat Obama, he said, “we need a fighter. We need a fighter, and someone who learned what America was about by growing up in communities just like this.”

In all, 10 states held primaries or caucuses on Tuesday, making it the most important day of the 2012 campaign so far.

Nationwide, exit polls revealed two broad trends about Republican voters. The first was that many remain un­enthused about the slate of candidates: In Ohio and Tennessee, more than four in 10 said they had cast their ballots “with reservations.”

The second was that GOP voters remain deeply split about what should be the top priority in this race: ideology or electability. In Tennessee, for instance, nearly four in 10 voters said that the ability to beat Obama is the most important candidate attribute. Romney wins that group.

But just as many said they are looking for a candidate who is a “true conservative” or someone with strong moral character. These voters picked Santorum by an overwhelming margin.

In Nolensville, Tenn., on Tuesday, Christi Lockwood, 40, a stay-at-home mother, said she realizes Santorum is the most conservative option. She’s not sure whether he would be able to attract independent voters in a general election.

But she said that wasn’t the point.

“I take the opportunity of the primary to vote for who I truly want to see as my candidate. And then, in the general election, I vote party,” she said. “You’ve got to let the process play itself out. I don’t like the talk about who is and who isn’t electable. The American people decide who’s electable, not the media.”

In Georgia, the meaning of Gingrich’s victory was simple: survival.

His momentum had bled away in the weeks after he won South Carolina on Jan. 21, and he had indicated that he would drop out of the race if he could not secure his home state.

Exit polls showed that Gingrich drew significant support from evangelical Christian voters, winning about half of the ballots from that bloc. He also captured more than half of the votes of strong tea party supporters, marking the first time that any candidate has taken a majority of any state’s tea party votes.

In Atlanta, Gingrich took the stage to wrestler Hulk Hogan’s theme song, whose lyrics are: “I am a real American. Fight for the rights of every man.” He then ran through a brief history of his campaign, counting all the times his candidacy had been declared dead — when other candidates had surpassed him as Romney’s main rival. Gingrich said he always came back: “It’s all right. There are lots of bunny rabbits to run through. I’m the tortoise.”

He told the crowd that Romney’s strategy of outspending his opponents would fail against Obama in the general election.

“What you have to have is somebody who knows what they believe, understands how to articulate it so that it cuts through all the media . . . and has the guts to take the president head-on every single time he’s wrong,” Gingrich said.

He promised to carry on: “In the morning, we are going on to Alabama. We’re going on to Mississippi. We’re going on to Kansas. And that’s just this week.”

As Super Tuesday got closer, the three top candidates made final pitches in the kinds of places that have defined their campaigns.

In Tennessee, Santorum found enthusiastic fans at the historic white clapboard Nolensville First United Methodist Church, about 20 miles southwest of Nashville.

“I think he’s the one who fears God the most,” said Chris Newell, 58, an insurance salesman from Nolensville. “I think he stands for conservative principles.”

In Ohio, Romney made an appearance Monday at a guardrail factory, one of many industrial sites he has visited to tout a back-to-work message. On Tuesday, he voted in his home town of Belmont, Mass., with his wife, Ann, before heading to a private family dinner. He told reporters afterward that he expected to stay up late watching the returns.

“I’m hoping for a good win here in Massachusetts, and what happens elsewhere I don’t know, but I think we’ll pick up a lot of delegates, and this is a process of gathering enough delegates to become the nominee, and I think we’re on the track to have that happen,” Romney said.

Gingrich, whose campaign has been marked by his fascination with space, made a final stop at . . . Space Camp. Visiting the U.S. Space and Rocket Center in Huntsville, Ala., he repeated his much-mocked call for a colony on the moon.

“Of all the places we’ve held rallies, this may be the most amazing,” Gingrich said, despite the comedic parodies his focus on space has generated — and the widely held view that the colony idea hastened his decline in the nominating battle.

“Far from backing off, I invite ‘Saturday Night Live’ to come to Huntsville to tape one of their skits,’’ he told a cheering crowd of a few hundred people. “They can tape it at the Space Camp. Because I want to restate: America has a destiny in space. That’s who we are.’’

Staff writers Philip Rucker, Felicia Sonmez, Rosalind Helderman, Amy Gardner and Krissah Thompson also contributed to this report.