Preparing for his maiden campaign outside of Asia, the reigning Japanese Formula 3 champion had been 20th on the opening day and seventh in Thursday's morning running.
Amid a flurry of afternoon qualifying simulations that saw times tumble by more than a second in the final 30 minutes, Matsushita recorded a 1m40.191s to top the timesheets.
He edged out Campos Racing's Arthur Pic by 0.159 seconds, while Sauber Formula 1 reserve driver Raffaele Marciello was fastest briefly before being bumped back to third.
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Sergey Sirotkin was fourth-fastest, ahead of Marco Sorensen, Stoffel Vandoorne, Daniel de Jong and Sergio Canamasas.
Earlier, Sirotkin topped the morning running, as the only driver to lap below the 1m42s barrier with a time of 1m41.797s.
In a session delayed by an overnight sandstorm, day one pacesetter Jordan King was second, ahead of Alex Lynn, Rio Haryanto, Pic and Vandoorne, though many chose not to run - or just completed minimal laps - in the conditions.
In the afternoon, Lynn was 17th and King 26th.
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