The Red Bull driver, who secured his fourth consecutive pole position at Suzuka on Saturday, led the race from start to finish to take his third victory at the Japanese circuit.
Championship rival Alonso retired from the race on the opening lap after making contact with Kimi Raikkonen.
Alonso's retirement means Vettel is now just four points behind with five races to go, and with a car that looks much more competitive than the Ferrari.
"Obviously it was an important step today but there's still a long way to go," said Vettel, the first man to score back-to-back victories this year.
"If you look at the last couple of races, after Spa it was very much up and down.
"I don't know what happened today, but at Spa Fernando was unlucky. You don't hope those things will happen to yourself but you know over the next few races it could happen. It is long season and crazy racing so far."
He added: "I am very careful. I think we had a long journey so far. It has been a tough year and there are still many races to go. Today I don't want to talk about the championship."
Vettel, who scored his third win of the season, said the weekend had been like a dream for him.
"I don't think it could be better," he added. "You come across these kinds of races very rarely.
"I had a very good race car, behaving very well, and it's very difficult to describe why because we didn't have major upgrades for this race. It's just that the car suited the track.
"It was a fantastic race and when you have a dream about how a race should be, that is exactly what you wish for. All in all it was fantastic."
Team-mate Mark Webber finished in ninth position after being sent into a spin by Romain Grosjean at the start.