Murray joins Cameron Norrie, Dan Evans and world doubles number one Joe Salisbury in the line-up for Britain's matches against the United States, Kazakhstan and the Netherlands at Glasgow's Emirates Arena.
Twenty-year-old Draper has surged from outside the top 250 to 55 in the rankings this season but Murray, who is eight places higher, gets the nod to provide singles competition to Wimbledon semi-finalist Norrie and 23rd ranked Evans.
A fifth player will be added later, giving Draper further chance to stake his claim along with doubles specialist Neal Skupski, who won his sixth title of the season with Dutch partner Wesley Koolhof in Montreal on Sunday.
Smith said: "I'm delighted to name such a strong team to take on the other three nations next month in Glasgow. We have really good strength and depth at the moment and selections have been tougher than ever.
"Emirates Arena has been a brilliant venue for our team in the past creating some of the best atmospheres and memories our players have experienced.
"We are very excited to bring Davis Cup back there for the first time in over four years and look forward to a packed-out stadium making some serious noise for our team."
Murray will play in the competition for the first time since 2019 having missed the most recent edition last November, when Britain lost to Germany in the quarter-finals.
He is guaranteed a hero's welcome at home, with Britain playing their matches on September 14, 16 and 18.
"It's always special playing in a Davis Cup tie in front of a home crowd," said the 35-year-old. "Some of the best moments of my career have come representing my country in the Davis Cup so to be a part of the team again means a lot.
"Obviously it's exciting that the competition is returning to Glasgow. We've had some brilliant results there in the past and this is another chance for us to create even more history. We've got a strong team and we'll be giving everything we can to get the win for the fans and book our place in the finals."
The latest incarnation of the historic competition sees four cities hosting groups of four teams each, with the top two in each pool progressing to a last-eight shoot-out in Malaga in November.