"I'm overwhelmed. There is a long road ahead of us but I’m absolutely over the moon. It feels like a long time since I’ve been sitting here,” said Hamilton, whose previous victory came in Belgium last August.
Hamilton tactically outsmarted the 23-year-old German who had pitted earlier and whose worn-out treads cost him speed in the closing laps.
"A big thumbs up to the guys back at factory, who put their heart into making the car what it is on the GP weekend. It feels amazing," said Hamilton.
Behind them, Mark Webber drove a brilliant race from 18th to third in the second Red Bull, only seven seconds behind Hamilton, after catching and despatching Mercedes' Michael Schumacher and Ferrari's Fernando Alonso, then doing the same to Ferrari's Felipe Massa, Nico Rosberg in the second Mercedes and McLaren's Jenson Button in the closing stages.
It was an odd race, that began with Button and Hamilton leading from Vettel, before Vettel repassed them as the first pit stops were made. He jumped Hamilton just before his stop on Lap 13, but admitted he was somewhat bemused when leader Button initially drove into the Red Bull pit before frantically being waved through to McLaren next door!
Further back, Sergio Perez was having another strong race as he made moves on Sebastian Buemi and Rubens Barrichello, but his enthusiasm got the better of him as he had a coming together with Nick Heidfeld which earned him a drive-through penalty, and then later drove Adrian Sutil off the track in to turn one. Paul di Resta had made a good start up to seventh place but complained of trouble with the rear of his car, and eventually slipped back to 11th behind Kamui Kobayashi.
Jaime Alguersuari was the only retirement from the race, his right rear wheel coming off immediately after a pit stop. It was a better day for Lotus though, as Heikki Kovalainen beat Pastor Maldonado's Williams to 16th on outright pace, while HRT also recorded its first double finish of the season.
1. Lewis Hamilton Britain McLaren-Mercedes 56laps 1hr 36m 58.226s
2. Sebastian Vettel Germany Red Bull-Renault +00m 05.1s
3. Mark Webber Australia Red Bull-Renault +00m 07.5s
4. Jenson Button Britain McLaren-Mercedes +00m 10.0s
5. Nico Rosberg Germany Mercedes-Mercedes +00m 13.4s
6. Felipe Massa Brazil Ferrari-Ferrari +00m 15.8s
7. Fernando Alonso Spain Ferrari-Ferrari +00m 30.6s
8. Michael Schumacher Germany Mercedes-Mercedes +00m 31.0s
9. Vitaly Petrov Russia Renault-Renault +00m 57.4s
10. Kamui Kobayashi Japan Sauber-Ferrari +01m 03.2s
11. Paul di Resta Britain Force India-Mercedes +01m 08.7s
12. Nick Heidfeld Germany Renault-Renault +01m 12.7s
13. Rubens Barrichello Brazil Williams-Cosworth +01m 30.1s
14. Sebastien Buemi Switzerland Toro Rosso-Ferrari +01m 30.6s
15. Adrian Sutil Germany Force India-Mercedes +1 lap
16. Heikki Kovalainen Finland Lotus-Renault +1 lap
17. Sergio Perez Mexico Sauber-Ferrari +1 lap
18. Pastor Maldonado Venezuela Williams-Cosworth +1 lap
19. Jarno Trulli Italy Lotus-Renault +1 lap
20. Jerome d'Ambrosio Belgium Virgin-Cosworth +2 laps
21. Timo Glock Germany Virgin-Cosworth +2 laps
22. Vitantonio Liuzzi Italy HRT-Cosworth +2 laps
23. Narain Karthikeyan India HRT-Cosworth +2 laps