The four-time winner at the Marina Bay street circuit complained to his team over the radio of an anti-rollbar issue but Ferrari, seeking their first win of the season, reacted too slowly and were unable to repair the car in time to set a fast lap.
Vettel was left stranded and will start Sunday's Grand Prix from the back of the grid.
The German, who is fourth in the driver standings after a poor season, felt he was kept out on the track too long by his team, shouting, "That was stupid. We're losing time" over the radio as he entered the pitlane after lapping slowest of the 22 drivers.
"Already in the first lap, going into turn one, something happened. Then in turn two, through turn five, I felt that the car was not the same," he told reporters.
"We don't really know what happened, initially I thought it was at the front but then it felt like the rear bar had broken. I hadn't felt anything wrong in (practice).
"Then I wanted to come into the pits but in the end we decided to stay out, and then we came in because it was hopeless anyway, we were just not quick enough, and there was no time within three or four minutes to fix the problem.
"For tomorrow, we have a long race in front of us, with a lot of safety cars. At least we have some new tyres, and even if it is not an ideal situation, we can still have a good race."
Vettel's team mate Kimi Raikkonen enjoyed a much better qualifying session, however, and the Finn will start in fifth place after lapping around a second behind Mercedes pole-sitter Nico Rosberg.
"Today we got more or less what we could from our car. Obviously it's a bit disappointing given where we finished, but this is a tricky place," he said.
"The race is going to be very long, and usually there's a lot happening, so we have to get the timings right and try to go forward."
(Editing by Neil Robinson)
