Draper beat Norwegian world number 313 Viktor Durasovic 6-2, 6-2 as he played his first competitive match since suffering a bruised bone in the summer.
The 24-year-old, now ranked 13 in the world, enjoyed a breakthrough season in 2025, when he won his first Masters titles in Indian Wells and broke into the top four, before the injury to his playing arm.
"It's been a really difficult time since I've been away from the sport. I knew I had to keep on practising my hardest and learn about myself," Draper said, according to the BBC.
"It's really good to play here in Oslo, to have my first match in a long time, to get a first point on the board for Great Britain."
Draper would have been expecting to face Norwegian top seed Casper Ruud, but the world number 12 pulled out of the tie following the birth of his first child last week.
Ruud's withdrawal left former junior No. 1 Nicolai Budkov Kjaer as the highest-ranked singles player in Norway's squad. The 19-year-old world number 133 went down 6-4, 6-4 to 26th-ranked Norrie
The tie concludes on Friday with Draper facing Budkov Kjaer and Norrie taking on Durasovic, as well as a doubles match.
The victors will face either Australia or Ecuador for a place in November's eight-team Davis Cup Finals.
Elsewhere this week, Serbia and Chile go head-to-head on Friday and Saturday in Santiago in a bid to set up a tie with Spain; qualifying second seeds Germany, without world number three Alexander Zverev, host Peru; and nine-time winners France take on Slovakia in a tie starting Saturday and Belgium, surprise semi-finalists last year, play Bulgaria in Plovdiv.
