Blackburn, a conservative who is also an ardent Trump backer, defeated former Tennessee Gov. Phil Bredesen, 74.
Cuomo's senior campaign adviser, Maggie Moran, said turnout was very strong, with numbers more closely resembling a presidential election than a midterm contest, and that benefited Democrats. "Certainly we feel good about where things are headed".
According to exit polling by CNN, two-thirds of voters said their vote in Tuesday's congressional election was about Trump, with more saying they were showing up at the polls to express opposition than support for the president.
A Democratic majority in the House would break the GOP's monopoly on power in Washington and give the party a check on Trump's agenda.
At the same time, Democrats re-elected embattled New Jersey Sen.
In suburban Virginia, Democrat Jennifer Wexton ousted incumbent Republican Barbara Comstock, while another Democrat, Donna Shalala, seized a Republican district in Florida.
A record number of women ran for office this election, many of them Democrats turned off by Trump's policy agenda.
He bet big on a xenophobic closing message, warning of an immigrant "invasion" that promised to spread violent crime and drugs across the nation.
Most Democratic candidates in tight races stayed away from harsh criticism of Trump during the campaign's final stretch, focusing instead on bread-and-butter issues like keeping down healthcare costs, maintaining insurance protections for people with pre-existing medical conditions and safeguarding the Social Security retirement and Medicare healthcare programs for senior citizens.
Those Republican candidates who run for office with Trump at the top of the ticket will have a hard time establishing their own political identities - and they may not want to, as U.S. presidents by and large win their second terms.
Districts with Large Foreign-Born Populations Sent Far-Left Democrats to Congress
In the first major race called Tuesday night, Lou Leon Guerrero was elected as Guam's governor, becoming the first woman to do so. Alexandria Ocasio-Cortez will be the youngest woman ever elected into Congress following her midterm election win Tuesday.
Democrats last had a majority in the House in 2010.
Colorado voters elected Jared Polis, making him the first openly gay man elected governor in the United States.
Mr Trump is expected to wield the axe in the wake of the midterms, with figures close to the president predicting that cabinet members will be forced out.
As polls begin to close, the White House is stressing the effort President Donald Trump put into a political ground game aimed at putting Republicans in the win column for Tuesday's midterm elections. Bernie Sanders of Vermont, Elizabeth Warren of Massachusetts, Sherrod Brown of Ohio, Sen.
And, as The New York Times pointed out, Mr Trump is counting on his negative campaigning to bring out his supporters.
"I'm convinced he has no idea what's about to happen: The fact that the House now has wide-ranging authority to investigate every inch of his administration".
Tuesday's elections also tested the strength of a Trump-era political realignment defined by evolving divisions among voters by race, gender, and especially education.
Republicans paved their path to victory by defeating Democrats Joe Donnelly of IN and Heidi Heitkamp of North Dakota. Hispanic voters favored Democratic House candidates by 33 percentage points - higher than the 18-percentage point gap that Democrats enjoyed in 2014, the poll found.
Since 1946, the president's party has lost an average of 25 House seats in midterm elections, according to Gallup.
While in Florida, young Democrat candidate Andrew Gillum was unable to turn the state blue.
The loss of the House is a serious blow to Trump who must now brace for the novel experience of oversight from Capitol Hill with Democratic committee chairmen promising constraints on his power that the GOP never attempted.