But, facing a rare whitewash as Hicks, the world No 33, took a 4-0 lead in just 48 minutes, Davis almost produced the comeback of his career.
In the end, the former British Under-19 champion was grateful to fall over the line as the three-times Grand Prix holder battled back to 4-3.
Meanwhile, Dave Harold gained revenge for a 13-2 defeat in last season's world championship by toppling Stephen Hendry to reach the semi-finals.
Harold claimed after a famous 5-2 victory: 'That's probably the worst I've played in the tournament.'
Hendry could not take advantage as his Stoke rival struggled near victory.
Harold might have won 5-1 but missed an easy pink in the sixth frame.