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CLEVELAND — Energized by his first primary victory, Gov. John Kasich of Ohio argued on Wednesday that he was the only candidate in the Republican field who could actually defeat the Democratic nominee in November.

“Let me tell you, neither Cruz nor Trump can win the general election,” Mr. Kasich said on NBC’s “Today” show, referring to Senator Ted Cruz of Texas and Donald J. Trump.

“You know, they can’t come into Ohio with the philosophies they have and win,” Mr. Kasich added. “If you can’t win Ohio, you can’t be president.”

Mr. Kasich kept his campaign alive by winning his home state on Tuesday, gaining 47 percent of the vote, compared with about 36 percent for Mr. Trump. When he gave his victory speech, he stood in front of a video screen that said, “As Ohio goes, so goes the nation.”

No Republican has ever won the presidency without winning Ohio in the general election. Mr. Kasich, who has been a notably moderate voice in the Republican field, likes to highlight his strength in a hypothetical matchup against Hillary Clinton, as several national polls have found him with an edge against Mrs. Clinton. Mr. Cruz and Mrs. Clinton have been roughly even in polls, while Mrs. Clinton has generally led in matchups with Mr. Trump.

In the “Today” interview, Mr. Kasich said that by prevailing in Ohio’s winner-take-all contest, in which he gained 66 delegates, he dealt Mr. Trump “a very, very big blow” as the New York billionaire tries to win enough delegates to secure the Republican nomination.

Mr. Kasich, who planned to campaign in Pennsylvania on Wednesday, said no candidate would win enough delegates to secure the nomination before the convention, which will be held in Cleveland in July. And he took a moment to bask in his victory.

“I beat Trump by 11 points,” he said. “I mean, that’s pretty remarkable, isn’t it? He lived in Ohio, he came in here, he trashed me and he lost.”