World number 16 Davis suffered a surprise 4-2 defeat against 79th-ranked Michael Wasley, while 49th-ranked Robertson was beaten 4-1 by 19th-ranked Joe Perry at the World Snooker Academy in Sheffield.
Things began well enough for Davis as he clinched the opening frame 79-7. But Wasley proceeded to reel off three in a row, winning the second 106-1 with a break of 66, the third 93-40 with a 62 and the fourth 68-25.
St Leonards-based Davis stayed alive by clinching the fifth 73-19 thanks largely to a 52 only for Wasley to seal a notable victory by rattling through the sixth 82-0 with all his points coming in a single visit.
Robertson, meanwhile, lost out in a match containing just a solitary half-century break. Perry took the opener 62-33 and doubled his advantage by claiming the second 70-32 aided by a 50.
Bexhill-based Robertson got himself on the scoreboard by notching the third 63-40, but Perry restored his two-frame cushion by edging the next 61-49 and completed the job in the next 67-6.
Robertson recovered from 3-2 down to edge out 76th-ranked Robbie Williams 4-3 in the previous round on Friday. Robertson won the first 71-23 only for Williams to grab the second 73-17 with a 60 and the third 68-13 via a 59.
A 64 gave Robertson the fourth frame 71-28 for 22, but Williams went back in front by capturing the fifth 76-20. Robertson nabbed the sixth by three points despite his opponent knocking in a 68 and won the decider 68-39.
Robertson, incidentally, had beaten both Williams and Perry in last month’s China Open Qualifiers. The venue stage of the Welsh Open is taking place in Newport this week.