In Davis Cup, Denmark and Spain are vying for a place among the world's eight best nations in the Final 8, which will be played in Italy in November.
In addition to Rune, Elmer Møller will be in action on Saturday against Jaume Munar, before it is finally decided on Sunday with a doubles match and two singles matches.
If Denmark wins, it will be the first time since 1988 that the Danish national tennis team is been among the world's eight best nations, albeit in a different tournament format.
The first set was challenging for Holger Rune, who came up against a highly motivated Carreño Busta.
The Spaniard was up with a break along the way, but Rune fought back in the Spanish heat on the gravel in Marbella.
While the Spaniard is relatively far from his best after a lengthy injury which has seen him drop in the world rankings while Rune has showed great form during the fall.
Rune's superior stability in his game also proved decisive as he took the first set 7-5.
Rune turned up the tempo a notch in the second set, and this initially caused the Spaniard to go a bit cold.
Just ten minutes passed before Rune led 3-0 as it seemed only a question of time before Carreño Busta would throw in the towel.
However, Carreño Busta responded well and broke back, and soon after restored parity at 3-3, all the while Rune briefly looked like a man who struggled to get his game going.
But Rune's problems were only short-lived, as the Dane played more or less flawlessly for the rest of the set and put Denmark in front in the highly anticipated clash between the two countries.