Republican Dan Sullivan claimed victory on Wednesday in a tight race for the U.S. Senate in Alaska, a week after elections gave Republicans control of the Senate and strengthened their grip on the House of Representatives.

Sullivan, a 49-year-old former state attorney general and natural resources commissioner, was leading his Democratic opponent, Senator Mark Begich, by 7,911 votes as of Tuesday night.

"From day one we told our supporters that we would run a campaign that Alaskans could be proud of and that's what we did," Sullivan said in claiming victory on Wednesday morning.

Alaska election officials tallied thousands of ballots on Tuesday cast by absentee and early voters, as well as people who voted at the wrong polling places. More votes will be counted in coming days, and officials said they hoped to certify the results by Nov. 28.

"There are tens of thousands of outstanding votes and Sen. Begich has heard from rural Alaskans that their votes deserve to be counted and their voices deserve to be heard," said Begich campaign manager Susanne Fleek-Green.

"He will honor those requests and will follow the Alaska Division of Elections as it continues its process and timetable to reach a final count," she added.

Begich, a former Anchorage mayor, had been slipping in polls since midsummer. His campaign touted his deep Alaska roots, where he was born and raised, while portraying Sullivan, who was born in Ohio, as an outsider.

Begich was narrowly elected in 2008 a few weeks after a jury convicted former Republican Sen. Ted Stevens on federal corruption charges, a conviction that was later set aside by a federal judge who cited prosecutorial misconduct.