The Russian fifth seed needed two hours, 14 minutes to see off the hard-working 103rd-ranked American 6-1, 6-7 (5/7), 6-0 with the roof closed on Rod Laver Arena on a wet Melbourne day.
Sharapova's 600th career win sets her up with a tough test next up against 12th seed Belinda Bencic, after the dangerous Swiss teen beat Ukraine's Kateryna Bondarenko.
"In a Grand Slam environment against anyone you can't expect to get away with it, and I didn't in the second set," she said, of allowing Davis back into the match in the second set.
"But overall really happy with how I came out in the third and stepped up, considering it's been, you know, many weeks since I've been in that position.
"So I was happy with the way I finished. I think I can take a lot of good things from that."
If Sharapova gets past Bencic she faces a potential quarter-final with defending champion Williams, who has proved unshakeable in every match they have played since 2004.
Sharapova's dismal record against the world number one was extended at Melbourne Park last year when the 21-time Grand Slam champion swept past Sharapova in the final.
To set up the mouth-watering showdown, she must get past the exciting Bencic, which won't be easy with the Swiss winning two tournaments last year -- including in Toronto, where she upset Williams.
"Yeah, an impressive player. A newer generation that's already established herself in the past season, beaten a lot of top players," Sharapova said of the 18-year-old, a product of the tennis school run Martina Hingis' mother Melanie Molitor.
"It's a tough fourth round. We haven't played against each other before, but I'm sure we'll be playing many times. It's great that we can start here."
- Dress for success -
Davis, just 5ft 2ins (1.57m) tall, had never been beyond the third round in four previous attempts at the season-opening Grand Slam, but she tried hard to fix that with a dogged second-set performance.
Sharapova, a full foot taller than the 22-year-old, had been in scintillating form so far, dropping just seven games in her opening two matches.
It went with serve until the fourth game when Davis fired a backhand long to give the Russian her first break point.
With the pressure on, the diminutive American couldn't cope with Sharapova's return serve and sent a wild forehand over the baseline to go 3-1 behind.
A winner here in 2008 and a three-time runner-up, the seasoned Sharapova, sweat dripping, held to love and then comfortably broke again.
The steely-eyed Russian was in no mood to give an inch and served for the set in just 26 minutes.
She continued to pressure the Davis serve and got a break in the third game of the second set.
But her concentration waned and Davis won her first break point of the match, with Sharapova's backhand into the net levelling it at 2-2.
Buoyed, the American stunned the fifth seed by breaking her again on a double fault.
Another break for the Russian hauled her back into the set and it went with serve till a tense tiebreak, which included an epic 27-shot rally. Davis finally won when Sharapova sent a forehand wide.
The Russian went for a break to compose herself, and refreshed and wearing a clean dress, she quickly broke to go 2-0 before powering across the finish line as Davis faded.
