A report in New Zealand claimed Quins are confident of signing Marshall due to a clause in his contract which would allow New Zealand's most-capped scrum half to leave Headingley if Leeds fail to stave off relegation.
However, Davies said: "He's got a contract here and we won't be going down. He's got a two-year contract and can be linked with whoever he likes - he's not leaving here."
Davies breathed a huge sigh of relief after his bottom-placed side's victory, which moves the Tykes to within two points of Bath.
Chris Bell's second-minute try got Leeds off to the best possible start and helped them establish a 16-6 interval lead.
Bath rallied to reduced the arrears to 16-14 after the break but the contribution of Leeds fly-half Gordon Ross proved decisive.
He not only converted Bell's early touchdown, but also notched five penalties and added a late drop goal to claim an impressive 20-point personal haul.
Olly Barkley's three penalties and Gareth Delve's 53rd-minute try were not enough for Bath.
Davies, whose side would have been 10 points adrift had they lost, said: "We had talked during the week about what we needed to do if things did get a bit tight or if we did go behind.
"We executed that on the field and I thought Gordon had one of his better games. He was very composed and kicked well at times.
"We've got 10 games to go and if we keep playing like that then we'll be picking up points."