The Red Bull and Mercedes touched on the run from the startline at Suzuka.
Vettel escaped with minor wing damage and went on to win, while Hamilton's left rear tyre went down at the first corner and caused sufficient damage to force his retirement.
"I had a very poor start, right between Romain [Grosjean] and Lewis, and clipped the front wing a bit," said Vettel.
"Lewis had a puncture after that. I couldn't go anywhere."
Hamilton said the contact was so light that at first he had no idea it had happened.
"I went for the gap, I felt like I had a clear gap, and then I was going into Turn 1 and the rear tyre was down," he said. "I can't believe my luck.
"I had probably the best start I've had all year. It was such a good feeling on the out-laps, the car felt really good. Really disappointing.
"It's such a long week, and when you don't even see the first lap, it's so gutting."
Although Hamilton made it back to the pits and rejoined, he said his Mercedes was so badly damaged that continuing would have been futile.
"The floor was destroyed," he said. "I was two seconds down per lap. The car was pulling to the right."