"I could not believe I was catching Vettel. He was putting up a great fight and it was great to get by him."
Hamilton closed to within 21 points of Vettel to be second in the drivers' championship and stop the German kicking off the season with a hat-trick of successes.
Both he and team-mate Jenson Button passed pole-sitter Vettel at the start. Red Bull elected to put their driver on a two-stop strategy in a bid to make up ground, but Hamilton found himself with much fresher rubber than Vettel and the two Ferraris in the latter stages, enabling him to overtake a series of drivers.
Reflecting with a degree of emotion on his 15th career success, Hamilton said: "This is one of the best GP wins I can remember. I got a great start and that gives you momentum.
"I knew the Red Bull had the real pace but the way it panned out with strategy and pit stops, I couldn't have asked for more.
"When you start you have to believe opportunities will come. I always think when I come into a race 'I can win this' ".
Start was key to victory - Hamilton
Hamilton performed his critical manoeuvre up the inside of a gentle left-hander, turn seven - not noted as an obvious spot for overtaking.
He said: "Sebastian is a great world champion. He is very good at defending, he has a great car, good traction and slightly better down-force. I had to take him at a place where he wouldn't expect it.
"You put so much into a season with preparations over winter. It feels like a long time since I won a GP (Belgium, August 2010). It feels like an eternity.
"My new option tyres seemed to last longer and the pit stops were fantastic. The car felt great and I was trying to nurse my tyres while picking up pace.
"The emotion comes from the desire to win and to be better and to compete against tough rivals. When you haven't won it feels like an eternity.
"The car has been great and we are having to push with everything we have to close the gap. We were smarter on strategy and on making it work."
“I was looking down and when I looked back up I could see I was in the wrong place”
Jenson Button explains the error on his first pit-stop
The result was a satisfactory one for McLaren as a whole, with Button's fourth place contributing 12 points to a team haul of 37, four more than Red Bull accrued.
That was despite a bizarre incident in which Button lost valuable time during his first pit stop when he momentarily parked his car between the Red Bull crew before being waved on to his correct position.
Button said of the incident: "I was looking down and when I looked back up I could see I was in the wrong place."
He went on: "Massive congrats to Lewis. It was one of those races with lots of overtaking, lots of action. I wasn't quick enough and really struggled to look after the tyres for some reason.
"It's enjoyable but also you're a sitting duck when people have new tyres. You try to fight back but you don't have the traction, it's tricky."
McLaren's chief operating officer Martin Whitmarsh added: "It's been a challenging weekend, so much emotion; there is so much action in the race that if you get it only slightly wrong you're toast.
"The first win of the year has been very emotional. You always worry about not winning a race in a season."
His team trail Red Bull by 20 points in the constructors' championship, with Ferrari 35 points shy of McLaren in third.
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