The president warned about the threat that North Korea's nuclear capabilities pose to international peace and security.
The president addressed the assembly as the eighth speaker during the morning session immediately after Slovakian President, Andrej Kiska, and long after the explosive speech made by US President, Donald Trump.
While addressing world leaders, the president backed the UN's call to halt the ethnic cleansing of Rohingya Muslims in the crisis-ridden country, warning that people are the primary victims.
He said, "The international community cannot remain silent and not condemn the horrendous suffering caused by what, from all indications is a state-backed programme of brutal depopulation of the Rohingya inhabited areas in Myanmar on the bases of ethnicity and religion.
"We fully endorse the call by the Secretary-General on the Government of Myanmar to order a halt to the ongoing ethnic cleansing and ensure the safe return of the displaced Rohingya to their homes in safety and dignity.
"In all these crises, the primary victims are the people, the most vulnerable being women and children."
The president also warned world leaders about the threat that North Korea's nuclear capabilities pose to international peace and security.
He said, "The most pressing threat to international peace and security today is the accelerated nuclear weapons development programme by North Korea. Since the Cuban missile crisis of 1962, we have never come so close to the threat of nuclear war as we have now.
"All necessary pressure and diplomatic efforts must be brought to bear on North Korea to accept peaceful resolution of the crisis. As Hiroshima and Nagasaki painfully remind us, if we fail, the catastrophic and devastating human loss and environmental degradation cannot be imagined.
"Nigeria proposes a strong UN delegation to urgently engage the North Korean Leader. The delegation, led by the Security Council, should include members from all the regions
"The crisis in the Korean peninsula underscores the urgency for all member states, guided by the spirit of enthroning a safer and more peaceful world, to ratify without delay the Treaty prohibiting nuclear weapons, which will be open for signature here tomorrow."
President Buhari concluded his speech by stating Nigeria's commitment to bringing global peace, security and development to the world in accordance with the principles and goals of the United Nations.
The annual debate of the UN General Assembly is an occasion for world leaders to gather to discuss global issues.
The theme of the general debate of the 72nd session is "Focusing on People - Striving for Peace and a Decent Life for All on a Sustainable Planet."
Other African leaders speaking during Tuesday's morning session are Guinea's Alpha Condé, Liberia's Ellen Johnson-Sirleaf, and Zambia's Edgar Chagwa Lungu.
Others set to address the assembly of world leaders during the afternoon session are Gambia's Adama Barrow, Mali's Ibrahim Boubacar Keita, Egypt's Abdel Fattah Al Sisi, Uganda's Yoweri Kaguta Museveni, Mauritania's Mohamed Ould Abdel Aziz and South Africa's Jacob Zuma.
President Buhari travelled to the general assembly with Governor Abdulaziz Yari of Zamfara state, Governor Dave Umahi of Ebonyi state, Governor Rotimi Akeredolu of Ondo state, Minister of Foreign Affairs, Geoffrey Onyeama, Minister of State for Environment, Ibrahim Jibrin, and the Minister of State for Budget and National Planning, Zainab Ahmed.