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.
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.
German Vettel, bidding for a fifth successive F1 title, was forced to pit quickly in the first session due to a heat problem and survived a skid on the grass in the second but his 51 laps for the day was a third of his total mileage for the winter.
"I'm quite surprised today, both Dan (Ricciardo) and I did a lot of laps so that's very good news, so let's see what tomorrow brings," Vettel said. "I learned a lot today, we're in a much happier place already."
Hamilton was left fuming when his car stopped at turn nine of his installation lap just five minutes into the first session.
With the vehicle hauled off by a tow-truck, the 29-year-old Briton walked back to the paddock without taking his helmet off and suffered the indignity of being stopped by security before retreating into his team's tent.
Brazilian Felipe Massa was 12th fastest for Williams, outshone in his first drive for the British-based team by Finnish team mate Valtteri Bottas, who was eighth quickest in the second practice.
Lotus, who missed the first test of the season because their car was not ready on time, suffered a black day with only French driver Romain Grosjean able to post a lap time, his best only 18th from 19 cars in the later session.
McLaren's Kevin Magnussen was the best of the three F1 rookies, the Dane ninth fastest in P2.
Kamui Kobayashi's return to F1 with Caterham was forgettable, as both he and his debutant team mate Marcus Ericsson were unable to post a lap time in either session due to reliability issues.
2nd Free Practice Session from the Formula One Australian Grand Prix at Albert Park on Friday
1. Lewis Hamilton (Britain) Mercedes 1:29.625
2. Nico Rosberg (Germany) Mercedes 1:29.782
3. Fernando Alonso (Spain) Ferrari 1:30.132
4. Sebastian Vettel (Germany) RedBull - Renault 1:30.381
5. Jenson Button (Britain) McLaren 1:30.510
6. Daniel Ricciardo (Australia) RedBull - Renault 1:30.538
7. Kimi Raikkonen (Finland) Ferrari 1:30.898
8. Valtteri Bottas (Finland) Williams-Mercedes 1:30.920
9. Kevin Magnussen (Denmark) McLaren 1:31.031
10. Nico Huelkenberg (Germany) Force India - Mercedes 1:31.054
11. Jean-Eric Vergne (France) Toro Rosso - Renault 1:31.060
12. Felipe Massa (Brazil) Williams-Mercedes 1:31.119
13. Sergio Perez (Mexico) Force India - Mercedes 1:31.283
14. Adrian Sutil (Germany) Sauber - Ferrari 1:32.355
15. Esteban Gutierrez (Mexico) Sauber - Ferrari 1:32.468
16. Danill Kvyat (Russia) Toro Rosso - Renault 1:32.495
17. Jules Bianchi (France) Marussia - Ferrari 1:33.486
18. Romain Grosjean (France) Lotus - Renault 1:33.646
19. Max Chilton (Britain) Marussia - Ferrari 1:34.757
20. Marcus Ericsson (Sweden) Caterham - Renault
21. Kamui Kobayashi (Japan) Caterham - Renault
22. Pastor Maldonado (Venezuela) Lotus - Renault
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- Sebastian Vettel
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- Nico Rosberg
- Lewis Hamilton
- Jenson Button