Norgaard hit 72 at The Belfry to finish on 16 under par, just two shots clear of South Africa's Thriston Lawrence.
His four-shot overnight lead was cut to a single stroke after he struggled to the turn in 37 while Lawrence birdied the 10th and 11th.
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.
But from just right of the green on the next hole, Norgaard miscued three straight chips 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.
Norgaard holed from 10 feet for birdie to remain two clear and a par on the 18th made him the third Danish player to win the British Masters after Thomas Bjorn in 2005 and Thorbjorn Olesen in 2022.
"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.
"I was trying not to think about it too much but this morning I almost threw up at breakfast, I was just so nervous.
"I'm just really bad at chipping sometimes, but luckily I made a few putts coming in and the way I bounced back I'm really proud."
Asked about winning the first Ryder Cup qualifying event ahead of the 2025 edition, Norgaard added: "I didn't even know. That's a good start!"
