Thorburn nicked the opening frame in a repeat of the 1983 world final, but Davis hit back by taking the next two with a break of 41 in the final frame proving crucial.
Elsewhere there were wins for Darren Morgan, Nigel Bond and Dene O'Kane.
Morgan was too strong against Joe Johnson as he won his two frames with room to spare; Bond had to come through two tough frames to beat Dennis Taylor 2-0; while O'Kane got the better of Barry West 2-0 too.
Morgan will now face Davis in what looks like the plum tie of the quarter-finals.
In the evening session, a clearance of 35 from Tony Chappel in the decider saw him edge out Tony Drago 2-1 and he will next face Alain Robidoux who defeated Les Dodd by the same score.
Karl Townsend produced a tournament high break of 86 but he still went down 2-1 to Tony Knowles who faces Karl Townsend next.
Jimmy White came out on top in the final match of the day, beating Mike Hallett 2-0, and will fight Dene O'Kane for a spot in the semi-finals.
The quarter-finals, semi-finals and final will all be played on Sunday.
All matches are best of three except for the final which is best of five.
Darren Morgan 2-0 Joe Johnson
Dennis Taylor 0-2 Nigel Bond
Dene O'Kane 2-0 Barry West
Steve Davis 2-1 Cliff Thorburn
Tony Drago 1-2 Tony Chappel
Karl Townsend 1-2 Tony Knowles
Alain Robidoux 2-1 Les Dodd
Mike Hallett 0-2 Jimmy White
Steve Davis v Darren Morgan
Dene O'Kane v Jimmy White
Alain Robidoux v Tony Chappel
Tony Knowles v Nigel Bond.