Davis, who made a 136 break against Indian billiards star Advani, will face Jamie Burnett on Friday for a place in the televised last 32 at the York Barbican.
Advani was a 6-5 winner over Davis's old rival Jimmy White on Wednesday.
Another star from yesteryear, James Wattana, went down by the odd frame to Ben Woollaston in the afternoon session.
In the morning, Jack Lisowski compiled a maximum break in winning his own third-round match 6-2 against Chen Zhe and will play Joe Perry in his last qualifier.
Mark Joyce was another heavy scorer in the first session with a 138 in a 6-4 victory over Andy Hicks while David Gilbert recorded a 127 and 124 in beating Tian Pengfei.
Dave Harold made a 130 in the afternoon but went down 6-4 to Cao Yupeng.
Qualifying Round Three results:
Mark Joyce 6-4 Andy Hicks
Steve Davis 6-5 Pankaj Advani
Jack Lisowski 6-2 Chen Zhe
David Gilbert 6-3 Tian Pengfei
Xiao Guodong 6-4 Adam Duffy
Liu Chuang 3-6 Luca Brecel
Ben Woollaston 6-5 James Wattana
Rory McLeod 6-3 Jimmy Robertson
Jamie Jones 3-6 Thepchaiya Un-Nooh
Dave Harold 4-6 Cao Yupeng
Anthony Hamilton 2-6 Ian Burns
Gerard Greene 4-6 Rod Lawler
Nigel Bond 5-6 Yu Delu
Liang Wenbo 6-5 Anthony McGill
Matthew Selt 6-4 Thanawat Tirapongpaiboon
Michael White 6-3 Craig Steadman