CNN's Jim Acosta claimed he had the upper hand in his faceoff with White House senior policy adviser Stephen Miller at Wednesday's daily press briefing.

"I think what you saw unfold in the briefing room is that he [Miller] really just couldn't take that kind of heat and exploded before our eyes," Acosta said in an appearance on CNN Wednesday night, hours after the face-off.

Miller spoke to members of the White House press corps about a revised bill from Sens. David Perdue of Georgia and Tom Cotton of Arkansas that would implement a merit-based point system for immigrants applying for legal permanent status. President Trump endorsed the immigration plan during a ceremony at the White House earlier Wednesday.

During the briefing, Acosta insinuated that the White House wants to allow only immigrants from Great Britain and Australia into the U.S. because they can speak English.

Miller then accused Acosta of having a "cosmopolitan bias" and said it was "insulting" that Acosta implied Perdue and Cotton's bill is racist.

"Have you honestly never met an immigrant from another country who speaks English outside of Great Britain and Australia? Is that your personal experience?" Miller asked. "It shows your cosmopolitan bias."

Acosta says Miller's characterization of him was ironic.

"It's not often you're accused of a cosmopolitan bias from someone who went to Duke University wearing cufflinks in the White House briefing room," Acosta said later on CNN.