But US media said the race was still too close to call, in a seat long held by the Republicans.
The election is seen as a key indicator ahead of November's mid-terms.
Tuesday also saw four states hold primaries to choose candidates for the mid-terms. It appears women will have a record number of nominees for both the House and governorship contests.
What has happened in Ohio?
Republican candidate Troy Balderson has been taking on Democrat Danny O'Connor in Ohio for a seat in the US House of Representatives. Ohio is one of the key battleground states in the mid-terms.
Mr Trump tweeted "Congratulations to Troy Balderson on a great win in Ohio. A very special and important race!"
But US news outlets have not called the contest yet. The New York Times gave Mr Balderson 50.2%, a lead of 1,754 votes in the 12th Congressional District race, with all precincts reporting. But at least 3,367 provisional ballots are still outstanding, which could lead Mr O'Connor to force a recount. He said: "We're not stopping now."
A provisional ballot is used to record a vote when there are questions about a voter's eligibility.
The BBC correspondent in Washington says even if the Democrats lose the seat, they can claim a moral victory in Ohio, and will point to the results as further evidence that they are poised for success in November.
Democrat House representative Ben Ray Luján said the seat should have been a "slam dunk" for the Republicans and the closeness of the contest was an "ominous sign" for them in November.
Mr Trump won the 12th Congressional District in 2016 by more than 11 percentage points and Republicans have held it since 1983.
Mr Balderson and Mr O'Connor will have to fight the seat again in November.
What about the primaries?
Kansas, Michigan, Missouri and Washington state have all been choosing candidates for the mid-term elections.
It appears that all the Republican candidates so far called as winners have been endorsed by Mr Trump.
No comments: