Vettel, victor of this season'so launching two races, completed eighth.
Behind the Ferraris it's a Mercedes second row, but it's Bottas starting third with Lewis Hamilton struggling, and starting tomorrow's race in fourth place.
The Finn had led the timesheets as the drivers embarked on their final flying laps and appeared set to clinch pole position after going fastest in the opening two sectors.
The 33-year-old, like Vettel a four-times champion, has taken an unprecedented five wins in Shanghai but recognised that his chances of a sixth had taken a knock.
On softer and fresher tyres after Red Bull made a strategic call to pit both its cars during the safety vehicle period, Verstappen was closing in on Vettel for the final podium place.
Sebastian Vettel has admitted to being impressed by Max Verstappen's apology for the incident that ended any hope of a podium finish for the German and resulted in a 10s penalty for the Red Bull star.
The Dutchman was pushing through the field and fighting Vettel for third place when he made a late move at the hairpin on lap 43 - clashing with his Ferrari rival and pitching both into a spin. Mercedes staying loyal to the softs with a longer first stint on the agenda for Sunday.
Vettel said he was not planning to defend his position hard, adding: "I didn't see him until very late". I don't think necessarily I need to be less aggressive or anything because it's nothing to do with being overly aggressive but maybe wanting too much.
"I mean he could have easily taken his front wing off, get a puncture, whatever, then it's game over for both of us".
Crucially, it was a full second clear of Hamilton's quickest effort. They were so fast on fresh tyres.