The West Yorkshire club, who won promotion to Yorkshire One last year and so now compete in the higher cup competition, will face Midlands side Silhillians next in an attempt to return to Twickenham.
They had too much pace for a big Aspatria side with winger Jack Marshall running in a hat-trick.
Replacement centre Danny Booth scored a brace and Keir Breakwell also got on the score sheet while they were awarded a penalty try.
Keir’s cousin Dale Breakwell made sure of the win with six conversions and two penalties.
Thornensians’ bid to make the semi-finals of the Senior Vase were dashed as they lost 10-7 at Tarleton, Tom Hughes’s try being in vain. But Goole were victorious in the National Junior Vase, beating Mossley Hill 24-0 to reach the semi finals.
Matthew Taylor scored two tries, while winger Tom Barker, centre James Doherty and full-back Danny Holt also scored as they booked a spot in the semi-finals against Long Eaton.
National Two North side Sheffield Tigers caused an upset, winning 38-14 at National One Hull Ionians in the Yorkshire Cup quarter-final on Saturday.
Both sides are at the wrong end of their respective divisions and need some momentum going into the final weeks of the season.
Ionians opened the scoring early, James Sanderson going over with Andy Humberstone adding the extras.
Tigers hit back with three tries to lead 19-7 at the break, Liam Wallace, Ryan Holmes and Pat Searle all on the scoresheet.
More of the same followed after the break, with Jamie Broadley and Pete Swatkins scoring, and Wallace grabbing his second. Greg Lound scored a late consolation try for the hosts, but it is the Tigers who advance to the semi-finals.
Hull secured a vital bonus point victory as they kept the pressure on Rossendale in National Three North, after a 27-19 win at Morley.
Wingers Ashley Smith and Steve Hihetah both went over early with Paul Hamber joining them on the scoresheet.
Morley fought back after the break, reducing the gap to a single point, before Lee Birch scored the bonus point try for Hull in additional time.
Bridlington survived a scare as they beat Pontefract 41-25 in the semi-final of the Yorkshire Shield.
The Yorkshire One league leaders needed extra time to see off Pontefract who sit third in Yorkshire Two.
The game ended all square at 20-20 but Bridlington’s fitness won out with Jamie Martin, Ben Dearing and Jimmy Thompson scoring in extra-time to seal the win.
Glen Boyd, Craig Fawcett and Liam Kay had crashed over for Pontefract with a brace from Callum Sanderson and another try for Martin building the tally for Bridlington.
Wath on Dearne will play in the Yorkshire Shield final after a closely-contested 15-10 win against Middlesbrough.
Wath took a half-time lead of 8-0 with Ashley Rothery’s try and Jack Whitlam’s penalty the difference.
Middlesbrough pulled three points back through a penalty before a Ben Cairns try and Whitlam’s conversion made it 15-3 to Wath. Middlesbrough did score a late converted try to put Wath under immense pressure, but Wath held firm.
A Josh Smith brace helped Skipton reach the semi-finals of the Yorkshire Silver Trophy as they beat Baildon 29-22.
Jamie Barker and Matty Langstaff also crashed over with Carl Paterson kicking the rest of the points. Baildon put in a battling second-half performance after trailing 29-5 at the break.
A strong showing from Old Otliensians saw them claim a 43-22 victory in their Yorkshire Silver Trophy quarter final at Aireborough.
The visitors’ forwards saw a lot of the ball, captain Sam Featherstone crediting a dominant performance from the pack and good retention from their own line-out.
Luke Cowdell scored twice, with Alex Murphy and Jack Smith among the scorers for Old Otliensians.