Verstappen made a hash of his final lap and Vettel lowered his time to 1:39.491, afterwards whooping and shouting to his team-mates in Italian as he celebrated pole.
"At the end of the day, to be second here, on this track, I think it is very promising", said 19-year-old Verstappen, who could have been the youngest driver in the history of the sport to start on pole.
Red Bull had topped the timesheets in every session leading up to qualifying and had looked on course for their first front row lockout since the U.S. Grand Prix in 2013, with Verstappen also fastest in the opening two phases of qualifying.
Championship leader Hamilton, who leads the German by three points with seven rounds remaining, qualified in fifth for Sunday's night race. "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".
Ricciardo suggested Red Bull could dominate the Singapore GP after his strong showing on Friday and still expects victory in Sunday's race, despite only qualifying third.
"I'm still confident we'll get victory".More news: Myanmar faces 'defining moment', must stop the violence
More news: 5 reasons why Manchester City will win the UEFA Champions League
More news: Nuclear Weapons Experts Warn Trump Against Abandoning Iran Deal
That in itself could be a tough task, with Ferrari's Kimi Raikkonen also ahead of the Briton and Singapore a notoriously hard circuit for overtaking. "It's important that we obviously keep that but I have a lot of faith we can stay there all weekend", he said.
We knew coming here it would be hard. I think we've got a good package.
But Verstappen was told to return to the pits slowly after reporting that the vehicle 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.
"You always go for it if there is a gap and it is possible", Verstappen added.
The Red Bulls headed out in a bid to respond, with Verstappen reclaiming top spot by just a fraction of a second. "So I needed to control the slides and the wheel slips more than I would wanted to and the pace is just slow", Bottas explained.