Amid widespread anxiety about their prospects at the polls next year, President Obama and Democrats found a rare kernel of hope in the off-year election results Tuesday night.

It came in Ohio, a perennially crucial battleground, where voters repealed a Republican-backed law curbing collective bargaining rights for many union members.

Across the country, the elections offered a mixed forecast heading into 2012, but party strategists say the results in Ohio may carry greater political importance.

“What happened in Ohio was meaningful,” said senior Obama strategist David Axelrod. “And the effort behind it was also meaningful. The sort of Democratic progressive machinery was sort of notched back there in the last election, and they rallied here in a big way.”

While Republicans picked up legislative seats in Virginia after trying to tie Democrats there to Obama, the GOP won fewer seats than predicted and appeared to have gained control of the state Senate only by a hair.

And in Mississippi, Democrats and their liberal allies scored an important victory, beating back an attempt to declare that a fertilized egg is a person under the law — a symbolic win for the left but one that is not expected to make such a conservative state competitive next year.

It was the Ohio win that Democrats hope will be most predictive. In it, they see evidence that their party can still prevail in an economically battered state and that, after being challenged by GOP governors in Ohio, Wisconsin and elsewhere, public-sector unions remain a potent political force.

“It is very clear that the headwinds we faced in 2010 are gone,” said Obama’s campaign manager, Jim Messina. “We are now to a place where we are organizing on the ground, and working hard and having some successes.”

But Tuesday’s results in Ohio also signaled at least one weakness for the president: Voters approved a state constitutional amendment to “preserve the freedom of Ohioans to choose their health care and health care coverage.” Known as Issue 3, the measure was seen as a rebuke of Obama’s health-care law. That was offset by the huge win on the collective-bargaining measure, known as Issue 2.

“If the Democrats claim a big victory and a testing of their organization on Issue 2, what happened on Issue 3?” asked Robert Bennett, former chairman of the Ohio Republican Party.

Still, Democrats saw the collective-bargaining vote as a test run for 2012. Volunteers from Obama’s campaign joined forces with labor unions for a full-blown push. Aides said that in the final two weeks of the campaign, the president’s network recruited more than 2,500 volunteers, set up about 30 staging areas across Ohio and reached voters through more than 250,000 phone calls and door knocks.

“This gave us a good chance to start the reelection campaign from a formal standpoint,” said Chris Redfern, chairman of the Ohio Democratic Party.

The collective-bargaining campaign was a major rallying point for the labor movement. The union-backed political committee formed to repeal the state law raised about $30 million for the effort.

But state labor officials cautioned that they don’t know whether Tuesday’s fervor will translate to Obama next year. The collective-bargaining law was complicated and hugely unpopular, and the public-sector unions that mobilized to repeal it were highly motivated — drawn more by the issue’s direct impact on their pocketbooks than by allegiance to Democratic Party politics.

Unlike in Wisconsin, where Gov. Scott Walker (R) pushed a similar law targeting public-worker bargaining but exempted public-safety workers, the Ohio provision applied to firefighters and police officers — a much more conservative constituency, and one that many voters believe has more right to collective bargaining.

“I think this was more focused on Issue 2. We had so much at stake,” said Mark Sanders, president of the Ohio Association of Professional Fire Fighters.

Sanders said the collective-bargaining issue will probably remain relevant next year. He said many of his colleagues — as well as police officers, many of whom consider themselves independents — will be more likely to vote Democratic if Republican candidates continue to support the rollback of bargaining rights.

“Regardless of political affiliation, [firefighters and police] are going to focus a little more on how elections affect us rather than some of those other off issues that don’t impact their career or their safety,” he said.

Stephen Loomis, a Republican who is president of the Cleveland Police Patrolmen’s Association, said Tuesday’s results reflected a victory for the political center — not one party or another.

He voted for Gov. John Kasich, but he said the move on collective bargaining by the new governor and his fellow Republicans reminded him of the push by Obama and the Democrats to revamp health care.

The presidential candidates “really have to start taking a step back and looking at marching to the middle,” Loomis said. “Those wackos on the left wing and those nutjobs on the right wing are the loudest, but this showed that we’re the most powerful.”

Staff writer Dan Balz contributed to this report.