Brazil and the Netherlands, whose citizens were among those executed, condemned the killings, and recalled their ambassadors to Indonesia for consultations in response.
The executions, carried out by firing squad shortly after midnight Sunday local time, were the first under Indonesia's new president Joko Widodo.
Citizens of Malawi, Nigeria and Vietnam were also executed, along with an Indonesian woman, officials said.
Brazi's President Dilma Rousseff issued a statement expressing indignation at the execution of Brazilian Marco Archer Cardoso Moreira, saying the incident "severely affected" relations between the two countries. She also offered sympathies to Moreira's family.
Rousseff "deeply regret(ted)" that her personal appeals for clemency, including one made via telephone, had been ignored by Widodo in proceeding with the execution, said the statement.
Moreira, 53, who was caught at Jakarta airport with 13 kilograms of cocaine in 2003, was the first of his countrymen ever to be legally executed abroad, Brazilian officials said.
Brazil has also been lobbying on behalf of another of its nationals facing execution in Indonesia, Rodrigo Gularte.
The Netherlands also condemned the execution of Dutch national Ang Kiem Soei, with Dutch Minister of Foreign Affairs Bert Koenders labeling the punishments "tragic."
"My sympathies go out to their families, for whom this brings a dramatic end to years of uncertainty," he said in a statement.
Dutch King Willem-Alexander and Prime Minister Mark Rutte had made overtures to the Indonesian president on the matter, which had been a recurring talking point in the relationship between the countries in recent years, he said.
"The government has used all possible means -- legal, diplomatic and political, at bilateral and European level -- to persuade the Indonesian authorities to cancel the execution," he said, adding he had temporarily recalled the Dutch ambassador in response.
"Every effort has been made, up to the very highest level."
Since entering office in October as a reformer, Widodo has drawn criticism from rights groups for his vocal support of the death penalty.
Last month, he publicly stated that he planned to reject the 64 appeals for clemency he had received from drug offenders on death row.
The hardline approach has raised concerns for other foreign nationals sentenced to death, including the two Australian ringleaders of the so-called "Bali Nine" ring, arrested in 2005 with more than 8 kg of heroin.
Indonesia, which has extremely strict drug laws, resumed executions in 2013 after a four year break, but no executions were carried out in 2014.
Amnesty International's research director for Southeast Asia and the Pacific, Rupert Abbott, said in a statement that the executions were "a seriously regressive move".
"The new administration has taken office on the back of promises to make human rights a priority, but the execution of six people flies in the face of these commitments," he said.