The away side were 43-4 when the impressive Tom Taylor (6-61) trapped Alviro Petersen lbw, but Ashwell Prince (97) and Steven Croft (70) put on 160.
Lancashire were 264 ahead when their ninth wicket fell, but Davies (89) and Jarvis (17 not out) added valuable runs to leave a far tougher target of 365.
Derbyshire then lost both openers cheaply to end the day on 27-2.
Wicketkeeper Davies struck nine fours and five sixes to reach his highest first-class score as he dominated the partnership with Jarvis.
Peter Siddle and Tom Bailey then took the wickets of openers Ben Slater and Billy Godleman with Derbyshire needing to make the highest score of the match to win.
Lancashire's Alex Davies: "A lead of 250, 260 is quite slender going into a full day tomorrow so we knew we had to hang around and help the team get into a good position.
"Those extra runs will give us a bit more scoreboard pressure. The pitch is still doing a bit laterally and will start to go up and down as well which will help us.
"I think the weather is set fair and if we bowl anything like (we can), we should be able to knock them over."
Derbyshire's Tom Taylor: "I think it's definitely possible (for Derbyshire to win), it's just how many big partnerships we can get and we've got to chip in.
"The ball has swung and it will swing again in the morning so it would be a great win but we fancy ourselves to try and get them tomorrow."

Royal Berkshire Fire Authority announced on Tuesday it would close Pangbourne station as part of plans to help it save £1.4m.
In the public consultation, the closure of Pangbourne station was the most popular option.
The closure of Wargrave station in Wokingham Borough was deferred for a year, but Pangbourne station in West Berkshire is to close permanently.
The Royal Berkshire Fire and Rescue Service said there was a possibility Wargrave station could close if funding was not found elsewhere and if the station, which is currently manned part-time by retained firefighters, did not increase its availability from 2.4% to 60%.
West Berkshire Conservative councillor Emma Webster - who sits on the fire authority, said: "I understand the concerns raised by residents and the local community.
"I think the most important thing that people need to remember is that residents across the whole of Berkshire will remain safe.
"If they need to dial 999 they will get the service they expect in a prompt and timely manner to assist them in their hour of need."

Timothy Geaney admitted taking or permitting the images to be taken between April 2004 and February 2014.
He was made the subject of a community payback order and put on the sex offenders register earlier this year.
He has now been removed from his profession's register following a General Teaching Council Scotland hearing.
In February, Dumfries Sheriff Court heard how most of the images he had downloaded were of young boys partially dressed in erotic poses with no sexual activity.
In addition to his payback order he was ordered to take part in a special programme for sex offenders.
A GTC Scotland hearing was told that the nature of the offence meant Geaney had "condoned and participated in an industry that violates and abuses children".
It heard that his conduct was "fundamentally incompatible" with being a registered teacher.
The panel found that Geaney had failed to uphold the standards expected of the profession.
It also noted that the offending behaviour had taken place over a lengthy period of time "demonstrating that there was a pattern of behaviour, as opposed to an isolated incident".
"Given the volume of images and the duration of the conduct, taken with the lack of evidence of remedial action, the panel was of the view that there was a strong likelihood of the offending behaviour recurring," it concluded.
The panel found him unfit to teach and removed him from the professional register.
Geaney, who was 69 at the time he was sentenced, was not present at the hearing.
He was prohibited from applying to re-register for a period of two years.
Lancashire's Alex Davies and Kyle Jarvis put on 100 for the 10th wicket to frustrate Derbyshire on day three.
[NEXT_CONCEPT]
Plans have been announced to close a fire station in Berkshire.
[NEXT_CONCEPT]
A Dumfries and Galloway music teacher found with more than 21,000 indecent images of children has been struck off.