Vettel cruised to his third victory of the season, and second in a row, to move him to within four points of Alonso at the top of the drivers' championship, after the Spaniard dramatically crashed out at the first corner after contact with Kimi Raikkonen.
Felipe Massa did his chances of staying with Ferrari the power of good with second place, and raised the spirits of the Italian team after Alonso's exit.
Kamui Kobayashi delighted the home crowd as he took a career-best finish third, becoming the first Japanese driver to finish on the podium here since Aguri Suzuki in 1990.
But it was another poor day for McLaren as Jenson Button and Lewis Hamilton finished fourth and fifth, although the latter moved to within 42 points of Alonso.
Raikkonen was sixth ahead of Nico Hulkenberg, Pastor Maldonado - scoring his first points since winning in Spain - Mark Webber and Daniel Ricciardo.
Webber had started second but his race was ruined as Romain Grosjean again played a part in a chaotic start.
While Vettel got the jump from pole, mayhem erupted behind him.
Alonso was edged to the outside on the run down to turn one, but clipped Raikkonen's front wing, running wide before spinning and coming to a halt in the middle of the track.
The drama was not over and Grosjean, who received a one-race ban after causing a pile-up at the start in Belgium, turned Webber around as the field negotiated turn two.
The Frenchman received a 10-second drive-through penalty for his trouble but that will be little consolation to Webber, whose chances of victory were well and truly wrecked.
Nico Rosberg also retired after a collision with Bruno Senna as the safety car made an appearance for the fourth straight race here, coming in after just one lap on track.
Vettel escaped into a lead he held for the rest of the afternoon, with Button the main beneficiary of an eventful first lap, moving up from eighth to third behind Vettel and Kobayashi's Sauber while Massa was fourth.
Hamilton briefly moved up to fifth after Sergio Perez's over-ambitious move on Massa at turn one, but the Mexican put one over on the man he will be replacing at McLaren next year with a strong move at the hairpin.
Hamilton had the last laugh though when Perez's effort to go around the outside left him in the gravel at the hairpin.
With Vettel streaking away the status quo at the front remained until the first stops.
But while Kobayashi and Button were among those to stop first, Massa was able to stay out for a vital few laps more and was able to get ahead of the pair at his first stop.
The only other move among the front-runners was made by Hamilton after the second round of stops.
The Mercedes-bound Briton had been behind Raikkonen since the start, but braved it out with the Finn as he rejoined from his stop to take what became an eventual fifth place.
The two battles during the closing stages saw Button try to close in on Kobayashi and the gap was down to 1.8 seconds with seven laps to go.
But the Briton could not get close enough, missing out by half a second, while Michael Schumacher, having started 23rd, could not overhaul 10th-placed Ricciardo.
But no one could match Vettel as he claimed the full set of pole position, fastest lap and race win, the 24th of his career.
The German has to be favourite to take the crown given Red Bull's exceptional pace this weekend, with the reported introduction of a Double DRS device giving them a further edge.
Massa's second place gave him his first podium since the 2010 Korean Grand Prix, while Kobayashi became the first Japanese driver on the podium since Takuma Sato at Indianapolis in 2004.
Vettel admitted it had taken him half the race to realise his principal title rival was out, but admitted his car was "a dream" to drive.
He said: "I saw there was a safety car at the very beginning, we had a very good start which was very important as behind me there was a crash.
"I saw a Ferrari had crashed, but I was not sure who it was but halfway through the race I looked at the (timing) tower to see where the others are and I saw the car that was still racing was Felipe, so I assumed Fernando was not racing any more.
"The guys have pushed so hard over the last couple of months and when you dream at night you dream about being able to drive a car like that, the balance was fantastic and I was enjoying every lap, which I think was why we had such a big gap to the others and I am very happy."