The Red Bull one-two came at the end of a dramatic race at Sepang.
World champion Lewis Hamilton was leading the course with 15 laps remaining but his Mercedes engine blew up and he was forced to retire.
A tearful Hamilton held his head in his hands after the malfunction which has dealt a blow to his chances of securing a third drivers' championship in a row.
The 31-year-old Briton is now 23 points behind Nico Rosberg who survived a spin on the first bend involving Sebastian Vettel and a 10-second penalty to take third place.
Ricciardo celebrated his first win of the season on the podium with a "shoey" - drinking champagne from his driving boot.
Verstappen, who turned 19 on Friday, celebrated his runners-up spot and his birthday in the same fashion.