Zephyr Teachout, a liberal law professor, shocked political insiders when she swiped a third of the Democratic vote from New York Gov. Andrew Cuomo in the 2014 primary, offering a preview of hard-left discontent that has bubbled over in this year’s presidential race.

Now Mrs. Teachout, a Vermont native and veteran of Howard Dean’s 2004 presidential campaign, is banking on name recognition and an endorsement from Sen. Bernard Sanders, the hard-charging progressive who started a political “revolution,” to put her over the edge in a tight House race against Republican nominee John Faso, a lawyer and former state assemblyman.

Rep. Christopher P. Gibson, a Republican, is retiring from the seat in New York’s 19th Congressional District, and Democrats see a chance to use the grass-roots unrest embodied by Mrs. Teachout to cut into Republicans’ sizable House majority.

Mr. Faso said Mrs. Teachout is an outsider of the worst kind: She moved into the district just last year.

“I have long experience both in and out of government in this district; she has none,” he said. “She’s a carpetbagger from Brooklyn.”

The Teachout campaign and the Democratic Congressional Campaign Committee, which is prioritizing the race in this cycle, said voters just want to know whether their elected leaders will stick up for them.

“I live in Dutchess County, which is a lot like the farming community I grew up in and I understand the issues facing the district,” Mrs. Teachout said. “But I think people really want to know that their Congress member will fight for them and will have their best interests at heart. If I’m lucky enough to be elected, I won’t answer to anyone other than the people of the 19th District.”

The drama is unfolding north of New York City, in a sprawling district that is mostly south of Albany and reaches from the Hudson River Valley to the Catskill Mountains.

Voter registration in the district is evenly split into thirds. As of April 1, the district had about 147,250 Democrats, 144,900 Republicans and an equal share of unaffiliated or third-party voters.

The district also runs the gamut socially, with blue-collar workers, eco-friendly liberals and city folks who migrated up the Hudson. It wraps in urban, suburban and rural communities, adding up to a wide-open race.

“This is going to be one of those tough-to-call elections, and both parties are investing heavily,” said Len Cutler, a politics professor at Siena College in Albany County, New York.

Republican presidential nominee Donald Trump and his Democratic rival, Hillary Clinton, both have high unfavorability ratings, so it is unclear whether either of them will tip the scales in the House race.

“I think [Mrs. Teachout] is doing her own thing, just as John is doing his own thing,” said Mr. Cutler, who noted that he worked alongside Mr. Faso in the 1980s when they were staff members in the New York State Senate.

Mr. Faso has said Mr. Trump doesn’t appear to be hurting or helping him.

“He certainly has strong supporters, and he has strong detractors,” he told Albany’s Talk 1300 radio host Fred Dicker on Thursday.

Mrs. Teachout is supporting Mrs. Clinton, even though she wrote a November piece in The Huffington Post that said the former secretary of state had “some corruption problems” because so many corporations had paid her speaking fees, donated to the Clinton Foundation or gave to her presidential campaign.

Indeed, Mrs. Teachout is following the grass-roots road map drawn by Mr. Sanders, the Vermont independent who pulled Mrs. Clinton to the left during a hard-fought presidential primary.

On the trail, Mrs. Teachout criticizes fracking, big money in politics and trade deals that “help multinational corporations make more money while shipping our jobs overseas.”

Voters, she said, “feel like no one is listening, and they’re right.”

“Our democracy is being drowned out by lobbyists and special interests who’ve rigged the system in their favor. That’s I’m running for Congress: to lift up people’s voices,” Mrs. Teachout said.

Mr. Faso, meanwhile, considers himself a centrist Republican like Mr. Gibson, the retiring congressman. He said he is focused on jobs and the economy back home, with an eye toward national security issues and reining in the debt in Washington.

“I’m very concerned about the direction of the country. I’m concerned we’re bankrupting the next generation,” said Mr. Faso, who served as minority leader of the New York State Assembly from 1998 to 2002 and was defeated by Eliot Spitzer in the 2006 governor’s race.

Mr. Faso grew up in Long Island, though he highlights his family’s 30-plus years in Kinderhook to argue that he is much more in touch with the district than Mrs. Teachout, who grew up in Vermont and built her name in politics as a web organizer for Mr. Dean when the former governor ran for president 12 years ago.

An associate professor at Fordham University in New York City, Mrs. Teachout supported the Occupy Wall Street movement and took on Mr. Cuomo in 2014 to protest corruption in Albany.

She did well among voters within the 19th District, grabbing solid majorities in each of the 11 counties it spans, though Mr. Faso insists those were just votes against Mr. Cuomo.

“She’s a radical leftist,” he said, “and I’m a center-right Republican whose views and values reflect a majority of people in the district.”

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