Davis, seeded No 2, trailed 6-3 overnight and lost the first two frames yesterday. He then put together a run of four successive frame wins and had a fine chance to level at 8-8. But at 26-0 he missed a red to the corner pocket, giving Hicks the chance to make a frame-winning 55 break.
In frame 17, Davis again made a good start and was leading 20-0 when he missed a straightforward black off its spot.
Hicks, 21, kept his cool to compile an 84 break, which finished on the yellow, to clinch a best of 25 frames second-round tie with the veteran, Willie Thorne, starting on Friday.
Davis had only previously been beaten in the first round at The Crucible three times - losing to Dennis Taylor on his debut in 1979, to Tony Knowles, who dethroned him in 1982, and to Peter Ebdon three years ago. The multi- millionaire won his six championships in Sheffield between 1981 and 1989 and has won 70 major titles in his career.
"Andy played wonderfully well and simply outplayed me," Davis said. "I really got close to pulling off what would have been an incredible steal but he took his chances very well."
Hicks, the world No 33 from Tavistock, was "elated". The left-hander said: "This has to be the greatest win of my career. I knew I could beat Steve because I knocked him out of the Grand Prix in Derby earlier this season, but to eliminate the six-times world champion on his home territory is something else.
"I was most nervous when I walked into the arena for the first time but once I won the first frame I was OK. I had a little think about the possibility of losing when he pulled back to 8-5 and again at 8-7 when I missed a great chance."
Thorne, making his 20th attempt to win the title, booked his place in the last 16 with a 10-6 victory over another rookie, Tai Pichit. Leading 5-4 overnight, Thorne had just a couple of hiccups before completing his expected victory over the world No 261 from Thailand.
"Now I would love to reach the semi-finals for the first time," the 41- year-old from Leicester said. "I have been coming here since 1977 and never yet reached the last four. But the way the draw is opening up this would be a good year to do it."
Pichit, a former world amateur champion, beat Alex Higgins on his way to The Crucible but never led again after losing a 2-0 advantage against Thorne.