Nadal lost the first set in a tie-break and looked set to follow defending champion David Ferrer out of the tournament, who lost earlier in the day to Ukraine's Ilya Marchenko, ranked number 94 in the world.
But the 14-times Grand Slam winner rallied after a difficult start to take the match 6-7, 6-3, 6-1.
"In the second and third sets, I played well, little mistakes and I was very positive," said Nadal.
He added: "I think I changed a little bit at the beginning of the second (set).
"I played with a little bit higher intensity and being a little bit more aggressive, going a little bit more for the shots.
"When I was able to make that happen, everything changed, no?"
The powerful Carreno Busta squandered three set points in the first, before finally winning the tie-break 7-5.
The world number 67 stretched his opponent at times and it looked as if Nadal, who won last week in Abu Dhabi, might exit the competition in the first round as he did last year.
But he recovered to give himself a second-round match against the "dangerous" Robin Haase in the second round on Wednesday.