Ferrari, who were way off the pace in Friday's practice sessions, stepped up their game on Saturday and Vettel will start at the front of the field on Sunday as he aims for a fifth win at Marina Bay.
The German four-time world champion, who is three points behind Mercedes ace Lewis Hamilton in the World Championship, clocked a quickest time of 1:39.491 in qualifying, nearly half-a-second faster than Verstappen.
Daniel Ricciardo, who topped both P1 and P2 for Red Bull, will start from third on the grid after managing to edge out the second Ferrari of Kimi Raikkonen, who will go from fourth on Sunday.
It has been a tough weekend for the Silver Arrows so far and Hamilton will have to take victory from fifth position whilst his team-mate, Valtteri Bottas, lines up next to him in sixth.
Speaking after clinching pole, Vettel said: "I don't know [where that time came from]. I struggled also this morning. I shouted as loud as the crowd. It was unbelievable.
"I love this track. The car was tricky but was getting better, better and better. So pleased we got it done.
"I need to calm down. The car was amazing. It’s an amazing track when you feel the car was coming together. I knew we had it in us but we had to get there."