The world number one was trailing 10-11 in the decider and found herself 0-40 down. But she came back from the brink to level and went on to take the third set 15-13.
"I'm just trying to change myself to not think that much about the points," said Halep .
"It's really tough when the opponent has match point. If we compare the old Simona with the new Simona ... I was a little bit stronger in that moment, for sure."
Halep, 26, has a reputation for losing her temper and folding at key moments of important matches. But she said her work on staying calm at such junctures appeared to be reaping rewards.
"You just go and hit without thinking," she said of the match points. "I did that and I think Lauren was a little bit tired at that moment. The pressure was on her also, and I was a little bit stronger in that moment, for sure."
The final set lasted two hours and 22 minutes, virtually double the time of the first two sets.
"I never played a third set so long. I'm almost dead," said Halep. "My muscles are gone. I don't feel my ankle any more. I’m almost dead."
Davis belied her ranking of 76 in the world. She matched Halep for athleticism and consistency from the base line.
"I have never played a match like that before where it went so long in the third set. We were both fighting our hearts out," said the 24-year-old.
"Every point was just super long. I got to the point where I was so tired I just told myself to swing and move and for the most part it was very effective."
She said being involved in such a mammoth tussle against a top ranked player would help her development.
"For sure, there are a ton of positives. There's a lot that I think I can work on. I'm just going to really focus on all that I did well and just how well I handled myself at this tournament. "