WASHINGTON (CNN) -- The Republican Party's message makeover may soon face a critical test as it considers the Violence Against Women Act, legislation that draws into sharp contrast the differences between the two parties on women's issues.

The Senate passed the measure on Wednesday, 78-22, and sent it to the House of Representatives, where proponents expect a fight that would test Republican efforts to embrace inclusiveness.

The Senate bill includes services, such as legal aid and shelter stays to domestic violence victims without regard to their immigration status or sexual orientation.

"Today, the Senate passed a strong bipartisan bill to reauthorize and strengthen the Violence Against Women Act. This important step shows what we can do when we come together across party lines to take up a just cause," President Barack Obama said in a statement.

"The bill passed by the Senate will help reduce homicides that occur from domestic violence, improve the criminal justice response to rape and sexual assault, address the high rates of dating violence experienced by young women, and provide justice to the most vulnerable among us," he said.

Obama signaled that the House should move quickly.

With the Senate passage, the pending House debate also opens the door to the types of thorny battles that stymied the bill last year and may have helped cost Republicans support among many women voters in the presidential race.

Those fights included issues like barring agencies that receive funding under the law from discriminating against gays and lesbians, allowing immigrants who face domestic violence to seek legal status, and giving tribal authorities new power to prosecute cases on Indian reservations.

Republican strategists are keenly aware that the party must broaden its appeal to women and Latinos.

Political experts said that action on the bill could help with that effort, if handled carefully. Though, the issue remains a challenge for some.

"All of the Republican women in the Senate are co-sponsoring the Senate bill. However, in the House, the women are more conservative and have been on board with the House position in opposition to this and in opposition to including protections for homosexuals," said Michele Swers, a Georgetown University American government professor.

Swers said the measure will especially test House Majority Leader Eric Cantor's "new more compassionate GOP message."

The Violence Against Women Act mostly provides support for organizations that serve domestic violence victims. Criminal prosecutions of abusers are generally the responsibility of local authorities, but the act stiffened sentences for stalking under federal law.

Supporters credit the Clinton-era act, which expired in 2011, with sharply reducing the number of lives lost to domestic violence.

Last year, the Senate passed a similar bill by a slightly less convincing margin, while the House approved its version on a mostly party line vote. But Congress failed to reach a compromise.

The Senate measure had more Republican support this time, meaning that it might get through both chambers this session, political experts said.

"This is not and never should be a partisan political issue," Republican Sen. Susan Collins of Maine said on the Senate floor last week.

But Florida Republican Marco Rubio, a potential 2016 presidential candidate, was one of a handful of senators who voted against bringing the measure to the floor. He has cited funding concerns.

Conservative groups are also closely watching how Republican lawmakers navigate the debate.

"In its current form, (the bill's) narrow focus ignores many of the proven causes of violence, is subject to waste, fraud, and abuse, and -- in some cases -- is harmful to the very victims it was intended to help," Christina Villegas, a visiting fellow at the conservative Independent Women's Forum wrote for the group.

Villegas said Congress should instead fight for "more accountable and effective programs" and should change the law to "serve all victims of abuse," regardless of gender or sexual-orientation.

Well-heeled conservative activist groups, FreedomWorks and Heritage Action, have voiced similar opposition. Heritage Action considers the bill a "key vote" when issuing its conservative scorecard.

The payoff could be big for Republicans, political experts say.

"It can drag them back into the debate but it is also an easy opportunity and a way for them to symbolically show that they're trying to be more inclusive," Jennifer Lawless, director of the Women and Politics Institute at American University said of the Violence Against Women Act votes.

CNN's Kate Bolduan, Dan Merica and Matt Smith contributed to this report.