Rep. James T. Broyhill, a 12-term congressman known in North Carolina as "Mr. Republican," and former governor Terry Sanford, a Democrat who has not held elective office in 21 years, easily won their parties' nominations yesterday for the Senate seat held by John P. East (R-N.C.), who is retiring for health reasons.

In Ohio, James A. Rhodes, another former governor trying to make a political comeback, defeated two Republican rivals who had said he is too old for the job. Rhodes, who will be 77 in September, is seeking an unprecedented fifth term. Gov. Richard F. Celeste (D), 48, who lost to Rhodes in 1978, is seeking a second four-year term.

And in Indiana, Jill Long, a Valparaiso city councilwoman and university professor, easily defeated a follower of extremist Lyndon H. LaRouche Jr. for the Democratic Senate nomination. Long, 33, will face Sen. Dan Quayle, 39, who was unopposed in the GOP primary.

The most bitter primary battle of the day was in North Carolina. Broyhill, 58, was attacked throughout the campaign by his opponent, David Funderburk, as not being a true conservative. Funderburk, 42, a former ambassador to Romania, was backed by Sen. Jesse Helms' National Congressional Club.

With 96 percent of the vote counted, Broyhill led Funderburk by 67 to 30 percent.

"I want to make it crystal clear that this Senate race will be a referendum on Ronald Reagan's leadership . . . ," Broyhill said in a victory statement last night. "I support President Reagan, and Terry Sanford doesn't."

Sanford characterized the upcoming general election race as "a campaign to put a different voice in Washington."

Broyhill, who comes from a prominent furniture-making family, turned the other cheek to the barrage of negative commercials in the primary campaign until its final week, when he ran ads accusing Funderburk of "cheap political tricks."

Helms remained publicly neutral in the contest, even though the leaders of his political organization had recruited Funderburk, mapped his strategy and written his television ads.

Helms and East, who endorsed Funderburk, are expected to lead a Republican unity effort on behalf of Broyhill.

Some Republicans, however, expect the National Congressional Club, which specializes in negative campaign ads and direct-mail fund-raising and serves as a kind of shadow Republican Party in North Carolina, will do little on Broyhill's behalf in the fall campaign for fear of strengthening the moderate wing of the state GOP. "They're either going to elect their own candidate or destroy ours," state Sen. Daniel R. Simpson, a Broyhill supporter, speculated in the last days of the campaign.

In the Democratic primary, Sanford, who retired last year as president of Duke University, scored an easy victory in the 10-candidate field. With 96 percent of the vote in, he had 60 percent, more than enough to avoid a runoff.

North Carolina voters also overwhelmingly voted against an advisory referendum asking whether they wanted a nuclear waste repository. And they rejected a referendum that would have moved statewide elections to odd-numbered years, a proposition advanced by Democratic leaders uncomfortable running on the ballot with their national party's ticket.

In a key House primary in the 10th District GOP race for Broyhill's seat, state Sen. T. Cass Ballenger defeated Rep. George S. Robinson, 53 to 43 percent, with all the vote counted.

In Ohio, with 83 percent of the precincts reporting, Rhodes had 47 percent of the vote, state Sen. Paul Gillmor 40 percent and state Sen. Paul Pfeifer 13 percent. Rhodes, who served as governor from 1962 to 1970 and from 1974 to 1982, needed to win only a plurality for the GOP nomination.

In Ohio's Senate primaries, Sen. John Glenn easily defeated Don Scott, a LaRouche candidate, for the Democratic nomination, while Rep. Thomas N. Kindness was unopposed for the Republican nomination.

In Indiana's 8th Congressional District, voters set the stage for a rematch of the closest congressional race of 1984, giving Rep. Frank McCloskey (D) the nomination over another LaRouche candidate. McCloskey will face Richard D. McIntyre (R), who ran unopposed. In 1984, McCloskey defeated McIntyre by four votes.