Spain, which also triumphed in 2000, '04, '08, '09 and 2011, swept the best-of-three series by winning both singles matches, rendering a potential deciding doubles match unneeded.
"Of course, I want to give credit to all of us, but he's been unbelievable what he has done".
When Auger-Aliassime hooked a backhand wide on match point Bautista Agut was hugged by Brugera before being embraced by his team mates on the Spanish bench.
His match against Shapovalov was his seventh match in four days and the past three nights he had played well past midnight, including a nerve-jangling doubles decider with Lopez against Britain in Saturday's semi-finals.
But Sunday felt like an authentic Davis Cup final, even if an appearance from Pique's pop singer wife Shakira added a little more glamour than usual.
"I think after you saw the success of Bianca (Andreescu), the whole country was talking about this".
A blistering forehand on the mosey train up the decisive spoil in the predominant train, which Nadal served out when Shapovalov hit lengthy.
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Nadal stormed through the first set of his match against Denis Shapovalov 6-3, but Shapovalov improved in the second set and took it to a tense tiebreak which Nadal won after 16 tense points 7-6 (7) to give the Spanish the sixth Davis Cup in their history before the final doubles tie.
Rafael Nadal gave all he had during the inaugural week-long Davis Cup Finals and his reward was a glorious victory for the hosts that meant so much.
The 31-year-old has shown incredible ability to return for the final and has climbed to the occasion with the support of a party in Madrid.
Bautista Agut, ranked No. 9 in the world, was playing his first match since the death of his father earlier this week.
Nadal had two World Championship Points at 6-4 up, but couldn't beat Shapovalov, two ridiculous points from Nadal's challenger seeing the second change-over just as level at six points all. He was 33-22 on tour this season.
Canada entered the tournament as the No. 13 seed, then took down No. 10 Italy, No. 6 United States, No. 9 Australia and No. 17 Russian Federation en route to the final.
Shapovalov crossed the break points in each of his first two games before the compatriot left-hander took a 4-2 lead with a successive winning back-to-back.
Roberto Bautista Agut delivered a big blow to Canada's hopes of capturing its first Davis Cup title.