"(He) asked Malaysia to ensure a fair trial, and to set free Doan Thi Huong", Voice of Vietnam radio reported after Mr Minh's call with Malaysian Foreign Minister Saifuddin Abdullah.
Huong's co-defendant, Indonesian Siti Aisyah, was freed on Monday after a Malaysian court dropped the charge against her. The call between the foreign ministers came as a surprise as Vietnam had refrained from publicly lobbying for her release until Tuesday.
Vietnamese citizen Doan Thi Huong, 30, is on trial for murder in Malaysia for the brazen Cold War-style killing of Kim Jong Nam, the half-brother of Kim Jong Un, at a busy Kuala Lumpur airport in 2017.
PTI reported Indonesian officials released Monday a letter sent by the country's justice minister to the Malaysian attorney-general, which reportedly sought her release, saying she had been "deceived".
Ms Doan was due to testify yesterday after months of delay but the trial was adjourned after her lawyer said she was "traumatised" to learn she now faced the charges alone.
But the two women have repeatedly said they believed they were taking part in a prank and had been tricked by North Korean secret agents.
Lawyers: Decision not to free Viet accused is discrimination
She is the only suspect in custody after the stunning decision to drop the case Monday against Indonesian Siti Aisyah. Huong's trial was postponed until April 1 after Doan's lawyer said she needed medical treatment. "It's so unfair.
"There is a law that allows charges to be withdrawn".
Salim said trial judge Azmi Arifin had ordered Huong to make her defence on Thursday if there was no response from the AG. "There was clear evidence and she (Aisyah) confessed", James Chin, a Malaysia specialist at the University of Tasmania, told AFP in a message.
Kim Jong Nam had lived in exile in Macau for several years before the killing, having fled his homeland after his half-brother became North Korea's leader in 2011 following their father's death. He had been living overseas for years but could have been seen as a threat to Kim Jong Un's rule.
Prosecutors did not give any reason for the remarkable retreat in their case against Aisyah, whose home government had lobbied hard for her release. "That was what happened", Press Trust of India (PTI) quoted Mahathir Mohamad as telling reporters in Kuala Lumpur. Pyongyang has denied the accusation.
Analysts said Aisyah's release was in part due to politics and the improved relations between Indonesia and Malaysia that have come since Mahathir Mohamad returned to the Malaysian premiership a year ago after the stunning election defeat of Najib Razak.