Up 2-0 in the second set and closing in on victory, the Russian's serve crumbled, allowing Davis to reel off five consecutive games and eventually capture a tense second set tiebreak.
Fifth seed 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.