Norgaard carded a closing 72 at The Belfry to finish 16 under par, two shots clear of South Africa's Thriston Lawrence.
The 32-year-old's four-shot overnight lead was cut to a single stroke after he struggled to the turn in 37, and then playing partner Lawrence birdied the 10th and 11th.
However, Norgaard responded superbly by making a birdie on the par-three 12th, where Lawrence missed the green and a par putt from inside three feet.
The two-shot swing left Norgaard three ahead and another long-range birdie on the 14th restored his overnight cushion, only for him to make a complete mess of the next.
From just right of the green on the par five, Norgaard duffed three straight chips - the last moving just a few inches - before two-putting from the fringe for a double-bogey seven.
That cut his lead in half and Lawrence looked set to get within a shot after leaving a long eagle putt on the 17th just two feet from the hole, but Norgaard bravely holed from 10 feet for birdie to remain two clear.
A par on the 18th made Norgaard the third Danish player to win the British Masters after Thomas Bjorn in 2005 and Thorbjorn Olesen in 2022.
Compatriot Rasmus Hojgaard charged through the field with a closing 65 to finish third on 12 under, with France's Jeong weon Ko another stroke back after a costly double-bogey six on the 18th.
Australia's Jason Scrivener finished two under par on 286, while compatriot David Micheluzzi was one under and Haydn Barron eight over.
"You have no idea how much it means," Norgaard said. "I've been dreaming of this since I was 10 years old.
"I've had kind of a long career, just moving slowly ahead and every year getting a little bit better. I've never won on the Challenge Tour, nothing like that, so to win here for a first win I feel very good."
Asked about winning the first Ryder Cup qualifying event, Norgaard added: "I didn't even know. That's a good start."
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