"In the fall, we are going to take back Terry Sanford's seat in the United States Senate for the people of North Carolina," Edwards told supporters Tuesday.
Sanford, also a former governor, died two weeks ago of cancer at age 80. He had served one term in the Senate before losing to Faircloth.
Edwards, 44, got 50 percent of the primary vote, more than enough to avoid a runoff. Faircloth, seeking a second term, got four out of every five votes cast in the GOP primary to easily beat two political unknowns.
Ohio and Indiana also held primaries Tuesday to choose nominees to succeed John Glenn and Dan Coats, two of the five senators not seeking re-election this year.
In Indiana, Fort Wayne Mayor Paul Helmke pulled out a narrow victory over attorneys John Price and Peter Rusthoven to win the GOP Senate nomination.
Helmke now faces an uphill battle against former Gov. Evan Bayh, who was uncontested in the Democratic primary and has raised $3 million for the campaign.
Bayh's father, Birch, held the same Senate seat until Dan Quayle beat him in the Reagan sweep of 1980. Coats, 54, who took over the seat in 1989 when Quayle became vice president, is retiring to return to the private sector.
Glenn, a 76-year-old Democrat, is leaving after four terms to go back to space aboard the shuttle Discovery.
In the race to succeed him, Republican Gov. George Voinovich, 61, faces Democrat Mary Boyle, a former Cuyahoga County commissioner. Boyle, 56, had no primary opposition, while Voinovich, barred from running for a third term, easily won the GOP primary against token opposition.
Neither party had a gubernatorial primary in Ohio. The nominees are Republican Secretary of State Bob Taft - the latest in a long political line of Tafts and the great-grandson of the late president - and Democrat Lee Fisher, former state attorney general.
Indiana and Ohio also had primaries to replace long-time Democratic congressmen who are retiring.
Democrats in southeastern Indiana's 9th District chose Baron Hill, a former state representative, to replace Lee Hamilton, who is quitting after 24 years. Republicans nominated Jean Leising, a former state senator who lost the last two elections to Hamilton.
In Ohio's 11th District, which covers Cleveland's east side and suburbs, Cuyahoga County prosecutor Stephanie Tubbs Jones won the Democratic nomination to replace Louis Stokes, the state's first and only black congressman, who is leaving after 30 years. The Republican nominee is talk show host James Hereford.
In the 6th District, Ohio Lt. Gov. Nancy Hollister won the GOP primary, thwarting millionaire businessman Frank Cremeans' bid to reclaim the seat he lost to Democratic Rep. Ted Strickland two yeas ago.
By Ned Kilkelly