Mr Abe described the 90-minute meeting in Trump Tower, New York, as "candid", with a "warm atmosphere".
Some of Mr Trump's campaign rhetoric cast doubt over long-standing US alliances, including with Japan.
The meeting was Mr Trump's first face-to-face with a world leader since winning the presidential election.
The US and Japan have been key allies since the end of World War Two, when the US helped Japan rebuild its economy.
The president-elect has vowed to scrap the Trans-Pacific Partnership trade deal, which Mr Abe strenuously supports as a means of countering China's growing economic strength.
The deal was approved by the Japanese parliament, despite the likelihood that it would be cancelled when Mr Trump takes office.
Mr Trump has also said Japan needs to pay more to maintain US troops on its soil, and has floated the idea that Japan and South Korea should develop their own nuclear weapons to counter the threat from North Korean missiles.
The meeting was reportedly arranged when Mr Abe rang the president-elect to congratulate him, mentioning that he would be passing through New York on the way to an Asia-Pacific trade summit in Peru.
Speaking after the meeting, Mr Abe said: "We were able to have a very candid talk over a substantial amount of time. We held it in a very warm atmosphere.
"I do believe that without confidence between the two nations the alliance would never function in the future and as the outcome of today's discussion I am convinced Mr Trump is a leader in whom I can have great confidence."
The Japanese leader gave few details of the meeting but added the two agreed to meet again for deeper talks on a wider range of issues.
Donald Trump's daughter Ivanka and her husband Jared Kushner, who have emerged as key advisers to Mr Trump since his surprise election victory, also met Mr Abe at Trump Tower.
Media speculation over her role has included the possibility of her becoming ambassador to Japan.
In the days since the election, Mr Trump has been speaking to dozens of world leaders as well as possible cabinet appointees from his home and office inside the Manhattan skyscraper.
In other developments:
Elsewhere, vice president-elect and Trump transition lead Mike Pence has said he is confident Donald Trump's administration can work with Democrats.
Emerging from meetings with Senate Democratic leader Chuck Schumer and House Democratic Leader Nancy Pelosi, Mr Pence said he looks "forward to finding ways that we can find common ground and move the country forward".
Meanwhile, US President Barack Obama has urged his successor to stand up to Russia if it deviates from US "values and international norms".
Speaking after talks with the German Chancellor Angela Merkel in Berlin, Mr Obama said he hoped Mr Trump would "not simply take a realpolitik approach" to dealing with Russia.
Japan's PM Shinzo Abe has said he has "great confidence" in US President-elect Donald Trump and he believes they can build a relationship of trust.