Great Britain's hopes of reaching the ATP Cup semi-finals were ended after Canada beat Germany to qualify as group winners instead.
Dan Evans and Jamie Murray's stunning comeback from a set and a break down helped Britain beat the United States and kept alive their hopes of progress.
Their 6-7 (3-7) 7-5 10-8 win over John Isner and Taylor Fritz in the decisive doubles clinched a 2-1 victory for GB.
But Canada eliminated Britain after winning their tie against Germany.
A victory for the Germans - who were already out after Britain's win - would have sent GB through to the last four.
World number 14 Denis Shapovalov put Canada in command by beating Jan-Lennard Struff in the first singles match.
Shapovalov won 7-6 (7-5) 4-6 6-3 to give team-mate Felix Auger-Aliassime the opportunity to seal Canada's progress by beating world number three Alexander Zverev.
Auger-Aliassime, ranked 11th, secured their spot with a 6-4 4-6 6-3 win.
In a tight Group C, all four teams went into their final matches with a chance of making the knockout round.
Evans had got GB off to a good start with a 6-4 7-6 (7-3) win over the big-serving Isner, with the British number two keeping up a perfect record in the group stage.
But number one Norrie lost 7-6 (7-4) 3-6 6-1 to Fritz to send the tie into a doubles decider.
When Evans and Murray went 4-2 down in the second set it had looked as if they would not be able to repeat their heroics of the opening tie, when the pair grabbed a surprise victory over Germany.
But they broke back immediately and eventually took the second set to level, continuing their momentum in the match tie-break, in which they won the opening six points to take charge.
A tense finish ensued, however, as the Americans rallied to come within a point of levelling at 9-8 before the Britons sealed victory.
"We were just hustling. We just kept saying, 'Just try to get the ball back in play, start the point', and tried to use our skills after that," Murray said.
"We played an amazing point to break back at 4-2, got a bit of momentum and played a great [match] tie-break, which is ultimately what it comes down to so many times."
Canada will face the Russian team in the semi-finals.
The Russians advanced when Daniil Medvedev and Roman Safiullin beat Matteo Berrettini and Jannik Sinner 7-5 4-6 10-5 in the doubles to seal a 2-1 win over last year's runners-up Italy in Group B.
Spain and Poland meet in the other last-four tie at the season-opening men's team event after booking their semi-final spots on Wednesday.