During a meeting on Friday, Kono asked Suu Kyi's government to allow humanitarian and media access to the affected area, the resettlement of returned refugees, and the implementation of recommendations made by former United Nations secretary-general Kofi Annan.
Asked about the admission on Wednesday by Myanmar's military that its security forces and Buddhist villagers killed 10 captured Rohingya during clashes past year, Suu Kyi stressed the importance of the rule of law and said the military will take responsibility.
Suu Kyi was speaking following a meeting in Myanmar's capital Naypyidaw at which Kono asked her to ensure the "safe and voluntary" resettlement of those who have fled, according to Japanese news agency Kyodo.
Kono arrived in Rakhine state on board a Myanmar military helicopter on Saturday as the first minister from a foreign country to visit the region.
More than 650,000 Rohingyas have fled to neighbouring Bangladesh to escape a brutal crackdown in which security forces have been accused of systematic abuses tantamount to ethnic cleansing.
Aung San Suu Kyi stressed the importance of the rule of law and said the military would take responsibility.
Burma's state-run media said on Wednesday that authorities have started the land work to construct buildings to accommodate returned refugees from Bangladesh in northern Rakhine, where refugees will be temporarily placed after their citizenship is scrutinised.
The Japanese foreign minister said Tokyo, one of Myanmar's biggest foreign aid donors, planned to give about $20 million for humanitarian support to Rohingya refugees, subject to parliamentary approval, Kyodo said.
Myanmar and Bangladesh have been discussing a plan to repatriate the refugees.
While Kono is visiting Myanmar, the Japanese government announced a grant of $3million (£2.2 m) to help facilitate the repatriation of the Rohingyas.