'I don't feel it's a catastrophe, she said, 'but I don't want to be going home this early'.
In a third-round match against Angelique Kerber that was hyped as being worthy of a final, only one player competed at a championship level.
Djokovic will next face Hyeon Chung, who took out a Zverev for the second time this week.
She sealed the win on her first match point when 76th-ranked Davis fired a forehand into the tramlines, as the hard-hitting American coughed up a mind-boggling 73rd unforced error. "I knew I wanted to use the medical timeout because I needed it, but at the same time, I knew it was nothing major that can potentially raise the question mark on whether I can continue playing".More news: Roma prepared for Dzeko exit, declares Monchi
"It's never easy to be on the losing end of things, be sitting in a press conference talking about a loss", Sharapova said. "I had a great 2016 and a year ago was tougher but everyone who knows me knows I never give up".
But, if all else fails her, Sharapova will always have her competitive instinct and she dug in to retrieve the break and give herself a foothold in the match.
However, during her week strokes, Sharapova was quick to follow up but not enough to beat the former World number one. At 13-13 when I took the game, I said it's my lucky number.
Kerber won that 2016 event, went on to win the 2016 US Open and ended that year as the No. 1 player in the world.More news: Sergio Aguero hat-trick guides Manchester City past Newcastle United
No. 17 Madison Keys was a 6-3, 6-4 victor over Ana Bogdan.
Her speed of foot and ability to return Sharapova's pace with interest left the Russian struggling from the outset. She had lost their only previous meeting, nearly five years ago, at Indian Wells 6-2, 6-0, and she had already won two three-setters this week, while Halep had looked comfortable without dropping a set.
"I'm nearly dead", the tenacious Romanian said after the incredible battle, in which Davis lost a toenail and Halep served for the match four times. "After this match I think tomorrow is gonna be worse, but I just push, push myself, and we will see what's gonna happen".
The Frenchwoman, who won two WTA titles previous year to send her rocketing up the rankings, proved too much for unseeded Aliaksandra Sasnovich of Belarus, winning 6-3, 5-7, 6-2 on Hisense Arena. "But looking at the overall picture, the overall beginning of this year, finishing the tournament, the first thing is that I'm healthy". And it won't have escaped Kerber's attention that she is the last remaining Grand Slam champion in the women's draw.More news: Seven-up for Real Madrid as Bale shines
Players turning in upsets in the heat were Taipei's Su-Wei Hsieh stunning No. 3 Garbine Muguruza 7-6 (1), 6-4, unheralded American Bernarda Pera defeating No. 9 Johanna Konta 6-4, 7-5, Sharapova ousting No. 14 Anastasija Sevastova 6-1, 7-6 (4), Japan's Naomi Osaka beating No. 16 Elena Vesnina 7-6 (4), 6-2, and Belarus' Aliaksandra Sasnovich topping No. 28 Mirjana Lucic-Baroni 6-3, 6-1.