The $2.7 billion defamation lawsuit against Fox News by the election technology company Smartmatic can move forward, a New York judge ruled on Tuesday. But the judge tossed out Smartmatic’s defamation claims against the Fox News host Jeanine Pirro and a network guest, Sidney Powell.

Smartmatic sued Rupert Murdoch’s cable news networks last year, along with several Fox hosts and guests. The lawsuit accused them of damaging the company by promoting a false narrative about the 2020 election: that Smartmatic and other voting systems companies tried to rig the race against President Donald J. Trump. Smartmatic later expanded its legal battle against disinformation to the right-wing media outlets Newsmax and One America News Network.

On Tuesday, Justice David B. Cohen of State Supreme Court in Manhattan said in a 61-page ruling that, “at a minimum, Fox News turned a blind eye to a litany of outrageous claims about plaintiffs, unprecedented in the history of American elections, so inherently improbable that it evinced a reckless disregard for the truth.”

He added, “At this nascent stage of the litigation, this court finds that plaintiffs have pleaded facts sufficient to allow a jury to infer that Fox News acted with actual malice.”

He also declined to dismiss Smartmatic claims against Maria Bartiromo, the Fox Business star, and Lou Dobbs, whose Fox Business show was a frequent clearinghouse for baseless theories of electoral fraud in the weeks after Mr. Trump’s defeat. Fox canceled Mr. Dobbs’s program last year, one day after Smartmatic sued.

Citing a legal technicality, Justice Cohen dismissed most of Smartmatic’s defamation claims against Rudolph W. Giuliani, who, appearing on Fox News as a legal representative for Mr. Trump, said the technology company had “tried-and-true methods for fixing elections,” among other false assertions. Even so, Justice Cohen said there was “substantial” evidence that Mr. Giuliani “acted with actual malice insofar as he evinced a reckless disregard for the truth” and ruled that Smartmatic could try again. The judge allowed another part of Smartmatic’s defamation case against Mr. Giuliani to go forward.

Fox News vowed a swift appeal.

“While we are gratified that Judge Cohen dismissed Smartmatic’s claims against Jeanine Pirro at this early stage, we still plan to appeal the ruling immediately,” the network said in a statement. The network added that it would “continue to litigate these baseless claims by filing a counterclaim for fees and costs” under New York’s anti-SLAPP (strategic lawsuit against public participation) statute, which is meant to quickly set aside lawsuits that may be intended to chill free speech.

Fox News said it would do so “to prevent the full-blown assault on the First Amendment which stands in stark contrast to the highest tradition of American journalism.”

In dismissing the claim against Ms. Pirro, Justice Cohen said that while she had asserted on her show that Democrats “stole votes,” she had not specifically blamed Smartmatic’s software.

A spokesman for Smartmatic did not reply to a request for comment.

Fox News is also battling a related $1.6 billion defamation lawsuit from Dominion Voting Systems, which has accused the channel of advancing lies that devastated its reputation and business. A Delaware judge rejected an attempt by Fox News to dismiss Dominion’s lawsuit in December.