Papandreou was born on the island of Chios, Greece, the son of the leading Greek liberal politician George Papandreou. His mother, born Zofia (Sofia) Mineyko, was half Polish. Before university, he attended Athens College a leading private school in Greece. He attended the National and Kapodistrian University of Athens from 1937 till 1938 when, during the quasi-Fascist Metaxas dictatorship, he was arrested for purported Trotskyism. Following representations by his father, he was allowed to leave for the US.

In 1942, Papandreou enrolled at Harvard University, where he completed a doctorate in economics. In 1943, he joined America's war effort and volunteered for the US Navy where he served as a nurse at Bethesda Hospital for war wounded, and became a United States citizen. He returned to Harvard in 1946 and served as a lecturer and associate professor until 1947. He then held professorships at the University of Minnesota, Northwestern University, the University of California, Berkeley (where he was chair of the Department of Economics), Stockholm University and York University in Toronto, Canada. In 1948, he entered into a relationship with University of Minnesota journalism student Margaret Chant. After Chant obtained a divorce and after his own divorce with Christina Rasia, his first wife, Papandreou and Chant were married in 1951. They had three sons and a daughter. Papandreou also had a daughter out of wedlock living in Sweden.

Papandreou returned to Greece in 1959, where he headed an economic development research program, by invitation of Prime Minister Konstantinos Karamanlis. In 1960, he was appointed Chairman of the Board of Directors and General Director of the Athens Economic Research Center, and Advisor to the Bank of Greece. In 1963, his father George Papandreou, head of the Center Union, became Prime Minister of Greece. Andreas became his chief economic advisor. He renounced his American citizenship and was elected to the Greek Parliament in the Greek legislative election, 1964. He immediately became Minister to the First Ministry of State (in effect, assistant Prime Minister).

Papandreou took publicly a neutral stand on the Cold War and wished for Greece to be more independent from the USA. He also criticized the massive presence of American military and intelligence in Greece, and sought to remove senior officers with "anti-democratic tendencies" from the Greek military. In 1965, while the "Aspida" conspiracy within the Army, alleged by the political opposition to involve Andreas personally, was being investigated, George Papandreou moved to fire the defense minister and assume the post himself. King Constantine refused to endorse this move and essentially forced George Papandreou's resignation. Greece entered a period of political polarisation and instability, which ended with the coup d'état of 21 April 1967. When the Greek Colonels led by Georgios Papadopoulos seized power in April 1967, Andreas was incarcerated while his father George Papandreou was put under house arrest. George Papandreou, already at advanced age, died in 1968. Under American pressure, the military regime released Andreas on condition that he leave the country. Papandreou then moved to Sweden with his wife, four children, and mother. There he accepted a post at Stockholm University. In Paris, while in exile, Andreas Papandreou formed an "anti-dictatorship organization", the Panhellenic Liberation Movement (PAK), and toured the world rallying opposition to the Greek military regime. Despite his former American citizenship and academic career in the United States, Papandreou held the Central Intelligence Agency responsible for the 1967 coup and became increasingly critical of the U.S. Government. In the early 1970s, during the latter phase of the dictatorship in Greece, Papandreou, along with most leading Greek politicians, in exile or in Greece, opposed the process of political normalisation attempted by Georgios Papadopoulos and his appointed PM, Spyros Markezinis. In August 6, 1974, Andreas Papandreou called an extraordinary meeting of the National Congress of PAK in Winterthur, Switzerland, which decided its dissolution without announcing it publicly.

Papandreou returned to Greece after the fall of the junta in 1974, during metapolitefsi, and formed a new "radical" party, the Panhellenic Socialist Movement, or PASOK. Most of his former PAK companions, as well as members of other anti-dictatorial groups such as the Democratic Defense joined in the new party. He also testified in the first of the Greek Junta Trials about the alleged involvement of the junta with the CIA. At that year's elections, PASOK received only 13.5% of the vote, but in 1977 it polled 25%, and Papandreou became Leader of the Opposition. At the 1981 elections, PASOK won a landslide victory over the conservative New Democracy Party, and Papandreou became Greece's first socialist Prime Minister.

In office, Papandreou backtracked from much of his campaign rhetoric and followed a more conventional approach. Greece did not withdraw from NATO, United States troops and military bases were not ordered out of Greece, and Greek membership in the European Economic Community continued. In domestic politics, Papandreou's government carried through sweeping reforms of social policy by expanding health care coverage (the "National Health System" was instituted), promoting state-subsidized tourism for lower-income families, and funding social establishments for the elderly. In a move strongly opposed by the Greek Orthodox Church, Papandreou introduced, for the first time in Greece, the process of civil marriage. Also, under PASOK, the Greek State appropriated real estate properties previously owned by the Church. Papandreou introduced various reforms in the administration and curriculum of the Greek educational system, allowing students to participate in the election process for their professors and deans in the university, and abolishing tenure.

Papandreou was comfortably re-elected in 1985 with 46% of the vote, but, in the years to follow, his premiership became increasingly clouded by controversy and scandal. In 1989, he divorced his wife Margaret Papandreou and married Dimitra Liani, while in the same year he was indicted by Parliament in connection with a $200M Bank of Crete embezzlement scandal, and was accused of facilitating the embezzlement by ordering state corporations to transfer their holdings to the Bank of Crete, where the interest was allegedly skimmed off to benefit PASOK, and possibly some of its highest functionaries. Following the many repercussions of the so-called Koskotas scandal, the 1989 elections produced a deadlock, leading to a prolonged political crisis. Papandreou's PASOK's won 40% of the popular vote, compared to the rival New Democracy's 46%, and, due to changes made in electoral law one year before the elections by the then reigning PASOK administration, New Democracy was not able to form a government. In the wake of three consecutive elections between 1989 and 1990, the New Democracy leader, Constantine Mitsotakis, eventually received sufficient support to form a government. In January 1992, Papandreou himself was cleared of any wrongdoing in the Koskotas scandal after a 7-6 vote in the specially convened High Court trial, ordered by the Greek parliament, with the support of both main parties, New Democracy and PASOK.

Papandreou confounded his critics by winning the next general elections of October 1993 ; however, his fragile health kept him from exercising firm political leadership. He was hospitalized with advanced heart disease and kidney failure on 20 November 1995 and finally retired from office on 16 January 1996. He died on 23 June 1996, with his funeral procession producing an outpouring of public emotion.