He made the statement in a meeting with the visiting Foreign Minister Mohammad Javad Zarif in the Japanese city of Yokohama.
Abe noted that Japan will be mindful of the intentions of Iran's leaders and will continue diplomatic efforts to ease regional tensions, NHK reported.
The already volatile Middle East region has been experiencing an escalation in recent months, following several attacks on oil tankers in the Persian Gulf, according to Financial Tribune.
Zarif welcomed the role of Japan's government in resolving the issues, stressing that Iran does not seek to increase tensions.
"We believe every country should be able to enjoy its rights under international law," he said.
Abe pointed to his meeting with the Leader of Islamic Revolution Ayatollah Seyyed Ali Khamenei in Tehran in June, saying he had been assured that Iran has no intention to produce or possess nuclear weapons.
Nevertheless, he expressed concern about Iran's plan to step up uranium enrichment early next month and urged the country to abide by the restrictions set in the 2015 nuclear deal.
All Moves Reversible
Scaling up uranium enrichment is the centerpiece of the reciprocal plan adopted by Iran a year after the United States pulled out of the nuclear agreement and reimposed sweeping sanctions on Tehran.
The plan involves gradually exceeding limits set in the nuclear accord. Iran has already implemented two stages and the third phase is due in early September, but it has repeatedly declared that all moves will be reversed once the economic damage of US sanctions is compensated by other parties.
Japanese Foreign Minister Taro Kono had also urged Iran to comply with the nuclear deal in his meeting with Zarif earlier on Tuesday.
"Based on Japan's stance of consistently supporting the nuclear agreement, I asked the Iranian side to observe the agreement, to go back immediately to the commitments of the agreement, and to refrain from taking measures that would infringe on the agreement,” he told reporters after the meeting, Reuters reported.
Prospective Meeting
Zarif later posted a tweet in Japanese, describing his meeting with Japanese authorities as constructive.
"Constructive consultations were held on further strengthening bilateral relations, alleviating tensions in the Persian Gulf and securing the benefits of the nuclear agreement for Iranians."
Zarif and Abe also agreed to maintain close communication and work to arrange a summit between Abe and Iranian President Hassan Rouhani on the sidelines of the United Nations General Assembly meeting next month in New York.
Tokyo has sought to broker dialogue between Washington and Tehran, with Abe traveling to Iran in June as the first sitting Japanese political leader to do so in over four decades.
The longtime foes are also at odds over the tanker attacks. The US blames the incidents on Iran and has called for an international maritime coalition to safeguard passage through the Strait of Hormuz.
Iran denies the accusations and has warned against any aggression against its interests in the Persian Gulf waters.
Although countries, including Britain and Australia, have announced their participation, Japan has remained hesitant out of concern that doing so would hurt its friendly ties with Iran.
Zarif added in his tweet that Malaysia will be the next stop of his Asian tour to pursue the Foreign Ministry's diplomatic initiative.