Ricciardo dominated the opening day of practice and Verstappen dislodged him at the top of the timesheets on Saturday, with the Dutchman going fastest in the opening two phases of qualifying.
Qualifying would suggest Hamilton is set to lose his slender three-point lead over Vettel in the championship on Sunday.
Verstappen, 19, had looked set to become the first teenager on pole but Vettel stormed through with two blistering laps in the final session to clinch it by three-tenths of a second. "The vehicle was tricky but it came alive and it got better and better as the night progressed, so really happy we got it done".
Qualifying has often been crucial in Singapore, where seven of the nine races so far have been won by the driver who started from pole.
Sergio Perez complained about having "no grip at all" on his second Q2 run and ended up 12th after being unable to improve on his first-run time.
The Ferrari driver took pole position while the world championship leader had to make do with fifth on the grid.
"We knew coming here it would be hard", Hamilton told Sky Sports F1.
"We were already holding on for dear life.I got everything out of all the laps".
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But Verstappen was told to return to the pits slowly after reporting that the auto was "shifting by itself" while Red Bull teammate Daniel Ricciardo clipped the wall, forcing him to back out of his final attempt a low-fuel run.
"I'm after a dominant weekend and so far this has been a good start", Ricciardo said.
"Then we'll see and try to take every opportunity there is".
"I think I did just not as much Sebastian", Verstappen conceded. "I threw the sink at it and squeezed every bit of it".
Vettel, who lost the lead to Hamilton at the previous race in Italy, is a four times victor in Singapore and the pole was also his fourth in the city state as well as the 49th of his Formula One career.
Speaking afterwards, Verstappen said: "It's a shame we couldn't put it on pole but we were quite close".