Ten expected to have faced the firing squad, including nationals from Pakistan, India and Zimbabwe as well as Indonesians, were not put to death but officials said they would be executed at a later stage.
Authorities did not give a reason for the reprieve, but the prison island where they were expected to be executed in outdoor clearings was hit by a major storm as the other sentences were carried out.
President Joko Widodo has defended dramatically ramping up the use of capital punishment, saying that Indonesia is fighting a war on drugs and traffickers must be heavily punished.
Noor Rachmad, deputy attorney general for general crimes, said the latest executions were "done not in order to take lives but to stop evil intentions, and the evil act of drug trafficking".
He added that "the rest (of the executions) will be carried out in stages", saying that the timings had not yet been decided.
"Any executions that are still to take place must be halted immediately. The injustice already done cannot be reversed, but there is still hope that it won't be compounded," he said.
The UN Secretary-General Ban Ki-moon and the European Union had also voiced opposition to the plan in recent days.
The executions, which happened at 12:45 am (1745 GMT Thursday), came after a day of frenetic activity, with distraught relatives travelling to Nusakambangan island to say farewells to their loved ones and ambulances carrying coffins over to the heavily guarded penal colony.
Eleweke's lawyer, Afif Abdul Qoyim, told AFP the execution should not have gone ahead as his client this week filed a legal appeal.
"When this process in not respected, that means that this is no longer a country that upholds the law, nor human rights," he said.
Two people whose cases had raised high-profile international concern among rights groups were not executed.
The first was Pakistani Zulfiqar Ali, whom rights groups say was beaten into confessing to the crime of heroin possession, leading to his 2005 death sentence.
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