Bottas, on soft tyres, had been the early pacesetter, but a mistake on Turn Two saw Hamilton usurp him as practice leader, before the Finn bounced back and finished top of the classification with a time of 1.42.742. Ferrari's Kimi Raikkonen was second.
Four-times world champion Vettel ended the session in 12th, 1.602 seconds behind Bottas, after withdrawing early due to "a loss in the hydraulic pressure circuit".
That set the stage for an unpredictable qualifying session.
It was a rare poor final practice performance and a potentially significant blow to Ferrari's hopes of matching Mercedes' four Grand Prix wins this season.
Hamilton, who is 12 points behind Vettel in the championship standings, finished four tenths of a second behind Bottas.
Despite improving on Friday's underwhelming performance, the British driver will be desperate to translate his improving form into qualifying after enjoying a practice sweep in Baku last year only to clip the barriers in qualifying and subsequently starting 10th on the grid.
Red Bull's Max Verstappen recorded the fastest time in both of Friday's sessions, but withdrew in sixth after a complete power failure due to a suspected hydraulic issue, with the driver having previously complained about engine problems.
Hamilton and Romain Grosjean of Haas were two of several drivers who rode their luck on the challenging Baku circuit's narrow bends, while Jolyon Palmer's misfortune continued as his car caught fire.
The British driver received a vote of confidence from Renault Managing Director Cyril Abiteboul despite crashing during Friday's session, but finished at the bottom of the standings here after completing just four laps.
(Reporting by Ed Dove, editing by Pritha Sarkar)
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