Alonso had qualified in ninth place, but was elevated to eighth after a penalty for Lotus' Romain Grosjean and then advanced another position - to the clean right-hand side of the grid - after Ferrari decided to incur a penalty for Felipe Massa by breaking his car's gearbox seal.
The move paid off, with Alonso moving up to fourth on the opening lap and taking third following Mark Webber's retirement, and keeps the title battle with Vettel alive until next Sunday's climactic race in Brazil.
"We know that our championship is alive thanks to the first laps," said Alonso.
"We always qualified around seventh or eighth and finish the first lap in the first three or four positions.
"And after that, the race becomes a bit easier if you are in leading group. Today we knew we had a good chance. I tried to overtake people at the first corner and then the pace improved and we were in the leading group.
"It was not possible to have the pace of these two guys (Lewis Hamilton and Vettel). They were too far ahead, but I had enough to keep the guys behind.
"This weekend it is like a victory for us. Losing just three points was something no one thought yesterday night or Friday night after practice. So we are happy to have another good Sunday and score good points again.
"It's now great to be going to Brazil still in with a chance of winning this championship."
Alonso has not won a race since the German Grand Prix in July and needs to finish at least third to have a chance of winning the title in Sao Paulo.
If he wins, he would need Vettel to finish outside the top four.
"It is something we have been fighting for during the whole year," he said.
"Only Seb is in a better position than us. We should be proud of ourselves, enjoy the race and see at the end what the outcome is."
Hamilton took a narrow victory over Vettel, after overtaking him while Vettel was dealing with backmarker Narain Karthikeyan.
"Overtaking Vettel wasn't easy ... It was actually quite tricky," he said after the race
"When I finally got close to him, I seemed to be catching him in the first sector and that was difficult.
"And then the backmarkers came into play, finally. How many times has it happened to me when I got caught out? (by backmarkers). But this time, it worked out and I knew that was going to be the lap.
"I turned the engine up to maximise the revs and went for the inside, but he defended. So, I went to the outside and he closed the door a bit so I was very lucky."
Hamilton, who won the last race on US soil in 2007 at Indianapolis, said: "I'm so happy, the fans have been amazing.
"It's been one of the best, if not the best, grands prix we've had all year. For me and my team this is so special.
"It's been a long time since I've had a win and I'm so proud of what they have done."
Vettel, whose second place finish secured the constructor's championship for Red Bull for a third consecutive year, admitted afterwards he was unhappy with being held up by the HRT driver.
"I wasn't too happy to send a nice big invitation to Lewis when I had to go through Karthikeyan," he said.
"He was right behind in the DRS zone. He took that opportunity, fair enough, down the straight and he passed me.
"I tried to defend, but I knew he would have so much more speed.
"I was obviously not too happy. Lewis had one chance and he took it. After that I tried to stay with him, but there wasn't much between us."
Vettel, driving his 100th Grand Prix, now has a 13-point gap over Alonso heading for Interlagos.
"The car is working well and good enough to fight for the win," he said.
"We've been quick in Brazil the last couple of years, and I have extended my lead in the championship, so we are in a good position."
Follow us @SprtNationalUAE