Correction: An earlier version of this story incorrectly reported that Rep. Tammy Baldwin is the only openly gay member of the House. She is the only openly lesbian member of the House. The story has been corrected.

Wisconsin Gov. Scott Walker, third from left, talks to his Cabinet and staff at the state Capitol Wednesday, a day after Walker beat Milwaukee Mayor Tom Barrett in a recall election. (Andy Manis/AP)

Buoyed by Gov. Scott Walker’s comfortable win in his recall election, Wisconsin Republicans claimed momentum Wednesday in their bid to seize a U.S. Senate seat that has been in Democratic hands since 1957.

The Senate race — already historic for its potential to elect the first openly gay senator — is now shaping up as the possible linchpin in the GOP effort to claim the chamber’s majority.

Following several months of slow-but-steady improvements for the Democratic effort to retain control of the Senate, Republicans said Wednesday that Walker’s seven-point margin of victory served notice that any of their potential nominees could defeat Rep. Tammy Baldwin, the likely Democratic nominee and the only openly lesbian member of the House.

“That strong a victory does take some of the wind out of the sails of Democrats,” said Sen. Ronald H. Johnson (Wis.), whose 2010 victory over a three-term incumbent gave the state its first GOP senator in 20 years.

Democrats were quick to reject this analysis, saying that the recall election was unique. More critical, Senate Democrats said, was President Obama’s strong standing among the Wisconsin electorate in exit polling.

“It’s an extremely different race,” said Sen. Patty Murray (Wash.), chairman of the Democratic Senatorial Campaign Committee, predicting that Baldwin’s record of fighting corporate interests would sell in the general election.

Independent political analysts tended to dismiss both sides as overreacting, suggesting that Tuesday’s results confirmed something that was already evident: that the Senate race in Wisconsin will be close.

“It is worth watching the Democratic base to see whether they go into November demoralized, or whether they rally for the president, Tammy Baldwin and Democratic House candidates,” said Jennifer Duffy, senior editor of the Cook Political Report.

That this Senate seat is such a battleground is a bit illogical. For 24 years, Sen. Herb Kohl (D), an heir to the Kohl’s department store chain and owner of the National Basketball Association’s Milwaukee Bucks, was an untouchable political figure who never faced a difficult reelection challenge. From 1957 until Kohl’s 1988 election, the legendary William Proxmire (D) held the seat — so comfortably that he refused to raise money for his last two reelection bids.

When Kohl, 77, announced his intention to retire at the end of this year, Wisconsin faced its first Senate race without an incumbent since Proxmire’s retirement.

Republicans believe they have a strong shot at claiming the seats of two other retiring Democrats, Sens. Kent Conrad (N.D.) and Ben Nelson (Neb.), while another incumbent, Sen. Claire McCaskill (D-Mo.), faces a difficult reelection effort.

That would leave them just one seat shy of the majority. However, Sen. Olympia J. Snowe (R-Maine) announced her retirement from a seat that is now likely to tip toward Democrats, and two other Republican incumbents, Sens. Scott Brown (Mass.) and Dean Heller (Nev.), are facing tough reelection battles.

By expanding the map, Democrats have given themselves breathing room, which is why Republicans have become so bullish on the Badger State. With just 47 seats, Republicans may need to win some combination of Wisconsin, Virginia and Montana to reach the majority.

The general-election campaign in Wisconsin will bring millions of dollars in ads from outside groups trying to influence the Senate race.

But first, Republicans must settle on a nominee. Tommy G. Thompson, who was a four-term governor, is the most well known of the GOP candidates, having also served in the George W. Bush administration as secretary of health and human services. However, he has not been on a ballot in Wisconsin since his 1998 reelection, and his opponents are criticizing his work at Akin Gump, a lobbying firm where he has served as a consultant.

Former congressman Mark Neumann has won support from tea party leaders such as Sen. Jim DeMint (R-S.C.). But Neumann has become something of a perennial loser in statewide races, beginning with his 1998 Senate bid and then his loss to Walker in the 2010 Republican gubernatorial primary.

Eric Hovde is a late entrant to the race, a Wisconsin native who recently returned home after a lucrative investment career, and he could be a wild card.

Walker has not endorsed a candidate in the contest, and it’s unclear whether he has a natural ally. Despite his tea party support now, he won the nomination in 2010 over Neumann and his tea party allies.

Republicans concede privately that none of those three candidates is perfect, but they stand ready to go after Baldwin, who was recently named the most liberal member of the House. Her Madison-based district is one of the most liberal in the nation, leading Republicans to question whether she’s ideologically out of touch with key swing regions such as Green Bay.

The National Republican Senatorial Committee, minutes after Walker was declared the winner Tuesday night, sent out a Web video that served mostly as an opening shot in the campaign against Baldwin. It included footage of her declaring at a rally that she is in favor of a “government takeover” of the health-care system and speaking to unions protesting Walker’s moves against public-sector labor unions.

Framing the race ahead, Baldwin’s campaign responded to the NRSC video Wednesday in an online appeal to donors, suggesting that she will embrace her role of taking on special interests in the general election.

“She has the backbone and guts to stand up to powerful interests in Washington and do the right thing for Wisconsin,” campaign manager Karin Johanson wrote.