With most of the votes counted in the state and less than 0.5% ahead, Mrs Clinton was declared the unofficial winner by one state official.
In the night's other primary contest Senator Bernie Sanders won in Oregon.
Front-runner Mrs Clinton is almost certain to secure the nomination in July, with a significant delegate lead.
Alison Lundergan Grimes, chairwoman of the Kentucky State Board of Elections, told CNN that unofficial results confirmed that Clinton would narrowly win the state's primary contest.
Shortly afterwards, Mrs Clinton tweeted: "We just won Kentucky! Thanks to everyone who turned out. We're always stronger united."
But Mr Sanders told a rally in California that he would fight on until the last vote is cast.
Analysis - Anthony Zurcher, BBC News, Washington
It was a narrow win in Kentucky, but Hillary Clinton will take it.
As far as the delegate count goes, it didn't really matter whether Mrs Clinton edged out Bernie Sanders or he ended up on top. But from a psychological standpoint - which is the only real battlefield left in this Democratic nomination race - it is probably a welcome boost.
Now Sanders supporters won't be able to tout much of a winning streak heading into the delegate-rich primary in California next month.
The case his campaign might make to the unbound super-delegates - who have largely endorsed Mrs Clinton and will almost certainly have to switch sides to give him the nomination - will be even less convincing.
With each state that ends short of a resounding win for Mr Sanders, Mrs Clinton takes one step closer to the nomination. And there aren't many states left on the board.
Given the turmoil in Nevada last weekend and Sanders team's assertions that Democrats are taking its supporters for granted, however, the party seems to be making little progress toward coming together to defeat Donald Trump in the autumn general election.
These Kentucky results aren't likely to change that, either.
Nevada chaos should worry Democrats
In the Republican race, Donald Trump won the party's only contest on Tuesday in Oregon, which was no surprise as he was the only candidate left in the race.
The Kentucky Democratic primary will award 60 delegates to go to the party's convention in Philadelphia while Oregon's primary will award 74.
Pressure is rising on Mr Sanders, a senator from Vermont who has historically been an independent, not a Democrat, to drop out of the race.
But he said he still has a path to the Democratic nomination.
Speaking at the California rally, Mr Sanders recognised his campaign's "steep hill to climb" but called for his supporters to remain hopeful and "take our fight into the convention" in July.
But senior party figures are pressing the Vermont senator to do more to bring his supporters into line, after some of them disrupted a state convention in Nevada last weekend.
Majority leader Harry Reid said Mr Sanders faced a "test of leadership".
And Democratic National Committee chairwoman Debbie Wasserman Schultz described Mr Sanders' response to the violence as "anything but acceptable".
Republican Donald Trump had a busy day on Tuesday. As well as winning in Oregon he:
struck a fundraising deal with the Republican National Committee said he would meet the North Korean leader to discuss Pyongyang's nuclear programme called for a renegotiation of the Paris climate agreement involving more than 170 countries released financial records claiming he holds $10bn (£6.9bn) in assets.