The 40-year-old wasted several opportunities in the deciding frame after clawing his way back to 4-4 having been 3-1 and 4-2 down.
Bond, playing with a new tip on his cue, started confidently with breaks of 39, 63 and 55, allowing Davis only 13 points in three of the first four frames.
But after the interval, Davis's run of 86 - the highest of the match - reduced his deficit to one frame only for Bond to extend his lead again with a 42 break.
Davis kept Bond in his seat for most of the next two frames to set up the decider which lasted 30 minutes and Bond's break of 21, including the last three reds, proved enough.
"I don't give a monkey's about my form as long as I'm winning. I misjudged one shot early in the ninth frame and it killed me," said Davis, the six- times world champion, following his first-round exit.
Bond's reward for defeating the 1991 runner-up is a quarter-final against world champion Ken Doherty tomorrow.