Paire, 29, betrayed his nervousness with six double faults during the opening set against the world number 21.
Carreno Busta led 4-2 and had two chances to take a 5-2 advantage and serve for the set. However Paire managed to claw his way back.
The Frenchman, ranked 54th in world, was again in trouble when serving at 3-5 down. He saw off two set points before holding to trail 4-5.
The Spaniard then fluffed his opportunity to collect the first set. He squandered a third set point before Paire broke back to level at 5-5.
From there it was one way traffic. Paire, emboldened by the partisan crowd at Villeneuve-d'Ascq in northern France, claimed the next two games to win the first set.
And against a mentally shattered opponent who was increasingly hampered by an injury to his left thigh, Paire reeled off the next five games.
Carreno Busta stopped the nine game roll to male it 1-5. But that was his last significant contribution to the scoreobard.
Paire, playing freely, fired winners from all angles and raced through the third set in 24 minutes to register the first point for his country in the best of five series.
"I am really happy to win my first match in the Davis Cup," Paire told the French TV Station France 4 after the match.
"It wasn't easy at first. That was obvious as it was my first match in this competition. But I came through and won the point and hopefully that will help the team to win the semi-final and get to the final."