The 2008 world champion could not finish a lap in the first session due to an oil pressure sensor problem, but topped the timesheets in the second with a lap of one minute 29.625 seconds on a glorious sunny day at Albert Park.
Champions Red Bull, who arrived under a cloud after problems throughout winter testing, also made a promising start and breathed a sigh of relief.
Quadruple world champion Sebastian Vettel finished an encouraging fourth on the timesheets, with his new team mate Daniel Ricciardo sixth behind McLaren's Jenson Button.
Hamilton's quickest lap edged out team mate Nico Rosberg by 0.157 seconds as the pace of the two Mercedes' drivers carried on from their strong showing in winter testing.
"I was trying to feel the car. I missed obviously P1 (first practice) which I felt like was a big back-step," Hamilton told reporters in the paddock.
"I'm satisfied with P2...Not the most perfect day but I'm really glad I got some run in P2 and just grateful I got some time to feel the car."
"I've still got a lot to improve on, a lot to find. Otherwise the balance doesn't feel too bad."
Teams struggled to come to terms with the new turbocharged hybrid engines during pre-season, and continued to battle glitches during practice, with drivers cooling their heels for long stretches in the garages, as technicians tinkered frantically.
Excursions onto the track saw plenty of over-steering at corners and skids into the grass as drivers fought to handle the new braking systems, extra torque and reduced downforce of the V6 power units.
Ferrari's two-time world champion Fernando Alonso handled conditions better than most, posting the quickest lap of one minute 31.840 seconds in the first session and finishing third fastest in the second.
His new team mate Kimi Raikkonen, returning to Ferrari after four seasons away, had a low-key start, posting the seventh fastest time in the second session after being rolled back to the garage due to an aborted start at the exit of the pit lane.


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