The Canaries make an instant return to the top flight following relegation to the Championship in the 2019-20 season.
Only Watford, eight points behind, can now deny them the second-tier title for a third time.
It is the earliest they have secured promotion since their first in 2004.
Saturday’s game against Bournemouth (20:00 BST) can edge them closer to securing the title with four games remaining after that.
The Canaries began the weekend needing a maximum of two points from their remaining five matches to secure promotion but the Bees’ goalless draw with Millwall and the Swans being held 2-2 by Wycombe meant they went up without having to kick a ball themselves.
Norwich lost 13 of their last 15 Premier League games last season as they returned to the Championship after just one season in the top flight.
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