Karl Rove was incredulous. His colleagues at the Fox News decision desk had called Ohio for President Obama, a move he insisted just a few minutes earlier would be premature.

“We’ve got to be careful about calling this when we have 991 votes separating the candidates,” he scolded them. “I’d be very cautious about intruding into this process.”

Silence settled over the set.

“That’s awkward,” Megyn Kelly said, trying to break the tension. She and the co-anchor Bret Baier then offered to broker something of a compromise. One of them would walk down the hall and interview the number crunchers on the decision desk about why they made their call.

And so ensued the most bizarre on-air encounter of election night: a network anchor interrogating Arnon Mishkin, a member of the Fox News decision team and a respected voting analyst, forcing him and a colleague to defend their news judgment against Mr. Rove, one of the most powerful Republican fund-raisers and strategists.