Lisowski, who beat Judd Trump in his opening match of the tournament, trailed Davis 5-2 and seemed on his way out of the tournament.
Davis had edged a couple of scrappy frames after the match reached 2-2, but Lisowski then turned on the style and justified his reputation as one of the game's brightest new talents with breaks of 80,71 and 81 to take the match.
"I felt more pressure today than I ever have before, because I had a good win against Judd and I wanted to make it count," said Lisowski.
"I threw away two frames to go 4-2 down but then I held myself together and made my best ever comeback. I played good snooker and I'm just so pleased to have won.
"When I played Judd we were both really attacking, but I knew today it wasn't going to be like that and I had to play more safety. I was pleased with that side of my game and I'm learning to mix it up more."
Mark Williams also came through a thriller in similar style, recovering from 4-2 down against Ali Carter with breaks of 136, 81 and 79 to make the last eight.
Mark Selby beat Ricky Walden 5-2, ensuring that he will regain the world number one spot from Trump, while Shaun Murphy also made it into the quarter-finals with a 5-3 win over Rory McLeod.
In the afternoon session, Neil Robertson roared past Mark Allen in fine style with a 5-1 thumping, the Australian rattling in breaks of 113 and 138 on his way to the win. Robertson has lost just one frame so far in the tournament.
Stephen Maguire also enjoyed a 5-1 win as he beat Barry Hawkins, but it was in far more scrappy fashion: Maguire's highest break was 50 as he came out on top of an error-strewn affair.
Stuart Bingham came out on top of a higher-quality match against Robert Milkins, with both players making centuries as Bingham won 5-3, while Marcus Campbell edged former world champion Graeme Dott in a final frame decider t o win 5-4.
Last 16 results:
Jack Lisowski 5-4 Mark Davis
Mark Williams 5-4 Ali Carter
Mark Selby 5-2 Ricky Walden
Shaun Murphy 5-3 Rory McLeod
Neil Robertson 5-1 Mark Allen
Stephen Maguire 5-1 Barry Hawkins
Stuart Bingham 5-3 Robert Milkins
Marcus Campbell 5-4 Graeme Dott
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