The two-times World Championships runner-up eased past Davis for the loss of just the one frame as he booked his place in the last 16 where he will face Jamie Burnett.
Burnett was also a comfortable winner his round five clash, beating Ryan Day 4-2 to ensure his progression to the next round.
They were joined by Judd Trump, a 4-2 winner over Craig Steadman, Allan Taylor and David Grace, who whitewashed Robbie Williams and Gareth Allen 4-0 respectively.
Mark Williams, however, was made to work hard for his place in round six, the two-times world champion sneaking past Rod Lawler 4-3.
Luca Brecel also found himself caught up in a tense battle before he got the better of Joe Perry 4-3 to continue his run.
But Jack Lisowski had no trouble in completing his way, dropping just one frame as he set up a match against Grace.
Robbie Williams 0-4 Allan Taylor
Steve Davis 1-4 Allister Carter
Jamie Burnett 4-2 Ryan Day
Gareth Allen 0-4 David Grace
Jack Lisowski 4-1 Matt Couch
Joe Perry 3-4 Luca Brecel
Craig Steadman 2-4 Judd Trump
Mark Williams 4-3 Rod Lawler