Pannir Selvam Pranthaman was arrested in 2014 for having 52 grams of heroin and sentenced to death three years later.
He is due to be hanged on Thursday, making him the fourth person to be put to death in the Southeast Asian country this year after two others were executed for drug-related crimes and one for murder.
Singapore anti-death penalty activist Kirsten Han said the Appeals Court allowed the stay of execution on the grounds that there is an ongoing constitutional challenge by other death row prisoners to a section under the drug law.
Although Pannir is not involved, he has said in his plea the case could potentially impact his conviction.
Another factor for the stay is Pannir's complaint to the Law Society of Singapore against his former lawyer, where he is the main witness, Han said.
The court in its judgement allowed his bid to mount another appeal against his conviction.
"This is a huge relief and very good news, but a stay of execution is not a complete stop," said Han.
"What Singapore needs now is an immediate moratorium on the use of the death penalty, with a view to full abolition. Pannir lives tomorrow, but as Singaporeans we must keep fighting to make sure no one is killed in our names ever again."
Singapore's strict laws mandate the death penalty for anyone caught carrying more than 15 grams of heroin and 500 grams of cannabis.
However, critics say the law only targets low-level traffickers and couriers.
Pannir's family, his lawyers and various rights groups noted that the Singapore court had found that he was merely a courier transporting the substance.
The 36-year-old has said he didn't know he was carrying drugs.
Still, the court had to hand down the death sentence after prosecutors refused to issue Pannir a certificate of substantive assistance, vouching that he helped their investigation, which would have spared him the noose, they said.
Malaysian lawmakers called on their government to intervene to halt Pannir's execution and have him extradited to facilitate further investigation.
The Malaysian government, which recently scrapped the mandatory death penalty, didn't respond to media queries.
"We oppose the use of the death penalty on drug mules who are themselves victims in the million-dollar transactions by syndicates that are roaming free," Malaysian lawmaker Ram Karpal Singh was quoted as saying by The Star newspaper.
Pannir's last-minute reprieve was hailed by dozens of supporters attending a candlelight vigil at a park in Singapore.
Holding placards and torchlights, they chanted "Justice for Pannir" and "Execution is not the solution."
Dozens others gathered outside the Singapore High Commission in Kuala Lumpur for a similar vigil, singing songs and reading excerpts of poems and letters written by Pannir.
Pannir, the third of six children, is the son of a church pastor who worked as a lorry driver to make ends meet, said another Singapore activist Kokila Annamalai.
He loves music and played in the church band growing up, she said.
In prison, Pannir has penned letters, songs and poems about the anguish of being on death row and hopes of a second chance, she said.
His family has shared some of his songs, which were sung by several Malaysian singers.
