It also meant De Minaur would take on Roberto Bautista Agut (No. 13) as the top-ranked players from each team, with Carreno Busta and Marcel Granollers then set to play John Peers and Luke Saville in the doubles tie.
Carreno Busta broke Millman in the third game of the first set, but the tough Australian broke back in the fourth as the Spaniards serve faltered.
But from then, Carreno Busta cracked the whip, convincingly overpowering Millman with his rocket forehands, continually pushing the Australian further and further behind the baseline.
Millman had no answer for that power, with Carreno Busta breaking a total of three times for the first set.
The Australian No. 2 tried to be more aggressive in the second set, taking more risks and attempting to restrict his opponents chances to tee off.
It worked with limited success, but only lasted until the fifth game of the set when Carreno Busta broke the Australian.
Carreno Busta looked like he was in the sort of form that has taken him to two US Open semi-finals, while Millman looked slightly off from his usual dogged self in his first outing of the season.
He was worked side to side at will by the relentless Spanish No. 3, and the Spaniard served out the match comfortably to give his country the early initiative.
On Wednesday Australia are set to face Greece, who are spearheaded by Stefanos Tsitsipas.
The top team from each group of the ATP Cup progress to the final eight, with the two next-best second-place sides also going into the next stage of the teams tournament, which is in its second year.
