Up 2-0 in the second set and closing in on victory, the Russian faltered on her serve, allowing Davis to reel off five consecutive games and eventually capture a tense second set tiebreak.
Fifth seeed Sharapova left the court for a lengthy comfort break between sets but re-emerged with a fresh outfit and a renewed focus, and proceeded to hammer the hapless American.
Bashing consecutive aces with her first two serves, Sharapova roared through without dropping a game, closing out the match when her 103rd-ranked opponent swiped a backhand long.
"I don't have a chat in the mirror, I think I get fined for that," Sharapova said courtside of her trip to the locker-room.
"I changed my dress and felt a little bit lighter which made me move better. But other than that, that's all I can share."
The win was 28-year-old Sharapova's 600th on tour and the five-time grand slam champion now faces an inter-generational battle against 18-year-old Bencic.
Long touted for big things, 12th seed Bencic was the first player into the round of 16 after fighting back from a set down to overhaul Ukrainian Kateryna Bondarenko 4-6 6-2 6-4 in the earlier match at Rod Laver Arena.
Easily outstripping her previous second round best at the tournament, Bencic also became the first Swiss woman to reach the fourth round at Melbourne Park since Martina Hingis and Patty Schnyder in 2007.
