Having gained pole position yesterday afternoon, it appeared during the early part of the race that Lewis Hamilton had every intention of stretching his advantage.
However, on lap eight, the Mercedes driver suffered a puncture on the Wellington Straight, which left him with half a lap to complete before he could return to the pits.
That handed the lead to Vettel, who had overtaken Rosberg just after the lights had gone out at the start of the race.
It seemed as though that the three-time world champion was going to cruise to victory, but with nine laps left to complete his race was ended by a suspected gearbox failure.
Hamilton's teammate Rosberg took full advantage of both problems as he negotiated the remaining laps to claim maximum points.
Home favourite Hamilton recovered from his earlier issues to finish fourth, with Kimi Raikkonen in fifth.
Result in full:
1. Nico Rosberg (Mercedes)
2. Mark Webber (Red Bull)
3. Fernando Alonso (Ferrari)
4. Lewis Hamilton (Mercedes)
5. Kimi Raikkonen (Lotus)
6. Felipe Massa (Ferrari)
7. Adrian Sutil (Force India)
8. Daniel Ricciardo (Toro Rosso)
9. Paul Di Resta (Force India)
10. Nico Hulkenberg (Sauber)
11. Pastor Maldonado (Williams)
12. Valtteri Bottas (Williams)
13. Jenson Button (McLaren)
14. Esteban Gutierrez (Sauber)
15. Charles Pic (Caterham)
16. Jules Bianchi (Marussia)
17. Max Chilton (Marussia)
18. Giedo van der Garde (Caterham)
Not classified: Romain Grosjean (Lotus), Sergio Perez (McLaren), Sebastian Vettel (Red Bull), Jean-Eric Vergne (Toro Rosso)