>> Trump tweets he won't attend White House correspondents' dinner

Perez was elected Saturday afternoon. He soon named Rep. Keith Ellison, D-Minnesota, to be the committee’s deputy chair.

“Congratulations to Thomas Perez, who has just been named Chairman of the DNC. I could not be happier for him, or for the Republican Party!” Trump wrote.

>> See the tweet here

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Perez quickly shot back.

"Call me Tom. And don't get too happy. @keithellison and I, and Democrats united across the country, will be your worst nightmare," Perez tweeted.

>> See his tweet here

Early Sunday, Trump weighed in again.

"The race for DNC Chairman was, of course, totally 'rigged.' Bernie's guy, like Bernie himself, never had a chance. Clinton demanded Perez!" Trump tweeted, referring to Sen. Bernie Sanders, I-Vermont, and former Secretary of State Hillary Clinton.

>> See the tweet here

– The Cox Media Group National Content Desk contributed to this report.