Laslo Djere gave Serbia the lead by beating Gabriel Diallo in the opening singles rubber before world No.13 Auger-Aliassime's clinical display levelled the tie and earned Canada the crucial point to seal their spot in the knockout stages.
With group stage matches being played in four European cities, Serbia went on to win the Group B tie 2-1 when Vasek Pospisil retired with a knee injury just three games into the match with partner Alexis Galarneau, handing victory to the pair of Kecmanovic and Filip Krajinovic.
"It feels great. I'm really happy. Super relieved," Auger-Aliassime told reporters. "To come out and play this way, get the win, it's very good for me personally. I'm just really happy for the team.
"I feel this whole event my level has been really good. I'm really happy where my game is at. It's very nice to see a smile on everybody's face, make the team happy and proud. That's what I came here to do."
Serbia's victory kept them in the hunt for a spot in the knockout stages of this year's competition, which will be held in Malaga from November 22nd-27th, with Spain also in the running to book the second spot from the group stage in Valencia.
With two teams from each of the four groups going through, victory for Spain against South Korea on Sunday would see the host nation qualify in top spot ahead of Canada.
In Group D, Botic van de Zandschulp beat world number 13 Taylor Fritz 6-4 7-6(3) for his third singles victory in as many ties to help the Netherlands secure the top spot with a 2-1 victory over the United States.
Both teams advanced to the last eight stage.
Australia and Germany have also booked their tickets to Malaga from Group C and the undefeated teams will clash on the final day of the Davis Cup Finals group stage on Sunday with the top spot at stake.
Last year's runners-up Croatia blanked Argentina 3-0 to keep alive their quarter-finals hope with Italy already progressing from Group A in Bologna.
Croatia are in second spot behind the hosts and their hopes of progressing will depend on the outcome of Sunday's clash between group leaders Italy and Sweden.
