"That this race is too close to call speaks volumes about Danny O'Connor's strength and Republicans' expensive, failed playbook", said Rep.
Democrats had hoped a victory by O'Connor in the district that has been represented by a Republican for more than three decades would signal a coming Blue Wave in November.
President Donald Trump campaigned for Balderson, arguing Republicans need to control Congress and casting the midterms as a referendum on himself.
The current governor - who took on his role in January after Gov. Sam Brownback left state politics to join the Trump administration - raised more money than Kobach, received the National Rifle Association's endorsement, committed to his pro-life views and earned support from Kansas political legend Bob Dole.
Trump took the district with 53 percent of the vote in 2016, just one-point below Mitt Romney's performance there in 2012.
The special election in OH coincided with with a series of primaries for Senate, House and gubernatorial nominations in Kansas, Missouri, Michigan and Washington.
Polling expert Nate Silver, founder of the statistics site FiveThirtyEight, said Balderson was "a heavy favourite" to win based on the votes counted. He called the race "very close".
In Michigan, State Attorney General Bill Schuette won the Republican nomination for governor on Tuesday night, defeating three other candidates working to fill the seat to be vacated by conservative Gov. Rick Snyder.
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The narrow margin will be little comfort for Republicans anxious about the prospect of a Democratic wave heading into the November 6 elections, and it is certain to encourage Democrats who performed dramatically better than expected in a district Republicans have represented since the early 1980s.
Trump didn't seem anxious about Kobach's prospects.
"If Democrats are winning any of these long-shots seats on Election Night, the House is probably already a lost cause for Republicans", the report notes.
Cory Bliss, executive director of the Congressional Leadership Fund, a Republican-aligned SuperPAC, said: "While we won tonight, this remains a very tough political environment and moving forward, we can not expect to win tough races when our candidate is being outraised". You will have a Big win in November and be a tremendous Governor for the Great State of MI.
O'Connor, 34, however, onstage at his campaign headquarters in the central OH district, said the race was a "tied ballgame", with a rematch due in November's regular congressional election for the seat, which opened up when GOP Rep. Debbie Stabenow, who ran unopposed in the Democratic primary.
Michigan Democratic gubernatorial candidate Abdul El-Sayed campaigns with support from New York Democrat candidate for Congress Alexandria Ocasio-Cortez at a rally on the campus of Wayne State University on July 28, 2018 in Detroit, Michigan.
There are dozens of Republican-held seats more favourable to Democrats across the country than Ohio's 12th district, a predominantly rural and suburban seat previously held by the state's moderate Republican Governor John Kasich.
O'Connor has dominated Balderson on the local airwaves.