The 2008 world champion, winner in steamy Malaysia last weekend ahead of championship leader Rosberg, completed 14 laps and was fastest around the Sakhir circuit with a best time of one minute 37.502 seconds.
Rosberg's time, on an day where the track temperature hovered around 33 degrees Celsius on a bright and breezy afternoon, was 0.231 slower.
The timesheets reinforced Mercedes' early-season superiority, with four cars powered by their engines in the top five.
Alonso, who has finished fourth in both races so far this season, provided some amusement when Ferrari accidentally sent him out with one soft tyre and three mediums on his car. The Spaniard stopped in the pit lane and was pushed back.
Three reserve drivers made debut appearances for the season, with Brazilian Felipe Nasr given track time at Williams, Dutchman Giedo van der Garde at Sauber and compatriot Robin Frijns at Caterham.
Second practice will be cooler and held under the newly-installed floodlights, with the race celebrating its 10th anniversary by switching from an afternoon to day-to-night format for the first time.
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