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Querrey's comeback victory lifted the Americans into a quarter-final tie in April at Boise, Idaho, against Novak Djokovic's Serbia after Thomaz Bellucci pulled Brazil level with a 2-6, 6-4, 6-7 (7/9), 6-4, 6-3 upset of John Isner.
"It's an amazing feeling. I'm so excited," said Querrey, who thanked US supporters who missed the start of American football's Super Bowl spectacle to stay and cheer him to victory.
Isner and Querrey, in their Davis Cup home debut, won singles matches on Friday to put the Americans on the brink of advancing.
But Marcelo Melo and Bruno Soares upset the World No. 1 doubles duo of Bob and Mike Bryan on Saturday to keep the Brazilians alive.
The Americans have not squandered a 2-0 lead to lose a Davis Cup tie since a 1994 semi-final loss in Sweden. But they came closer than they wanted Sunday.
"Relieved is the first word that comes to mind," US captain Jim Courier said. "This was a really important match for Sam, I think, in his development. We had a couple stumbles but we ended up getting through the tape in the end."
Alves broke 20th-ranked Querrey twice to capture the opening set but the American bounced back to take the second, captured the third on a service winner and broke Alves in the fifth game of the fourth set to seize command.
But Alves smacked a forehand winner to save a match point in the 10th game and Querrey, who fired 26 aces in the match, netted a forehand and an overhead smash to surrender the break and set up the final tie-breaker.
Alves sent a forehand long and netted a backhand on his serve to hand Querrey a 6-3 lead and the American made the most of his second match point, ending matters when Alves sent a forehand beyond the baseline.
"I was just trying to stay positive and make him work on those points in the tie-breaker and it worked," Querrey said.
The US squad, which has won a record 32 Davis Cup titles but taken the trophy only once in the past 17 years, must face Serbia next.
"They are going to be a very difficult ask for us," Courier said. "But we will be ready to go."
It took six break-point chances in the eighth game of the final set, but 36th-ranked Bellucci finally cracked Isner when the American netted a slice backhand. Bellucci then held at love, his 21st held service game in a row.
"I think he was a little bit tired in the end of the last set," Bellucci said. "I was just trying to fight because he was making mistakes.
"For me it's very special. Isner is a great player. Of course, he was favored. It's always special to beat him. I'm very happy. He has a big serve. It's not easy to play against him on this type of (hardcourt) surface."
Isner, ranked 16th, suffered his fifth loss in a row in five-set matches.
"Today was extremely disappointing for me," Isner said. "You can't sugarcoat it. My five-set record is atrocious. It falls on me 100 percent. I've got to get better. I feel bad I didn't come through for my team."