It was Davis who came out on top, thanks to a break of 54 in the ninth frame, and he will play either Ding Junhui or Dave Harold in the last 16 on Thursday.
In the eighth frame, Stevens looked to have Hendry in real trouble but an excellent break of 50 put the Scot clear and he held his nerve in the final frame.
And both think the cuts should be strung out over a longer time frame to give the economy a break, particularly at a time when the Bank of England has seemingly run up against the limits of its ability to lessen the pain.
Hendry suggested he might make it a similarly tight finish when he won the 13th frame, after several misses, and then showed his class with a break of 116 - his only century of the match - to narrow the gap to 8-6.