By Agence France-Presse
"We were able to have a very candid talk over a substantial amount of time. We held it in a very warm atmosphere," Abe told reporters after their meeting at the billionaire's Manhattan skyscraper in New York.
"Without confidence between the two nations, our alliance would never function in the future," he said.
"As an outcome of today's discussions, I am convinced Mr Trump is a leader in whom I can have great confidence."
Abe thanked Trump for meeting him as he was shaping his cabinet ahead of his inauguration as US president in January.
"Despite his extremely busy schedule because of decisions to be made vis-a-vis appointments or other related business, he was kind enough to receive me," the Japanese leader said.
"I renewed my conviction that together with Mr Trump I will be able to establish a relationship of trust."
The Japanese leader gave few details of the meeting and only said he "conveyed my basic views" on various issues.
But he added the two had agreed to meet again for deeper talks on a wider range of issues.
Japan is one of Washington's closest allies but Trump alarmed Tokyo policymakers during the campaign by musing about pulling the thousands of US troops from the region and suggesting that officially pacifist Japan may need nuclear weapons.
Trump also vowed during the election to tear up the Trans-Pacific Partnership, a proposed vast trade pact backed by outgoing President Barack Obama and which Abe had made a top priority.