Victory over Mark King on Tuesday meant Davis, 52, became the oldest winner of a Crucible match since Eddie Charlton beat Cliff Thorburn in 1989.
Ahead of his clash with the world reigning champion from Wishaw, Davis said: "I shall be going in to try to win it. I've made my mind up, I'm fed up of making up the numbers, so I'm here to try to win."
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Hide AdIf Higgins – world champion in 1998, 2007 and last year – wins, he could face Neil Robertson in the quarter-finals after the Australian beat Fergal O'Brien 10-5. Robertson meets Martin Gould next.
Two former world champions concluded their first-round showdown as Scotland's Graeme Dott beat England's Peter Ebdon 10-5.
China's Ding Junhui crushed Stuart Pettman 10-1, while world No.6 Ryan Day lost 10-8 to Mark Davis, who will face Northern Ireland's Mark Allen next.
Paisley's Stephen Maguire opened up a 6-3 lead over Stephen Lee, and 2005 champion Shaun Murphy developed an 8-1 advantage over Gerard Greene.