He says their resignation would be "for the good of our community, our party and our candidates."
He said the federal corruption probe of the county, with the prime targets of Dimora and Russo, "has compromised the public trust."
Democratic officeholders, including Gov. Ted Strickland, County Commissioners Tim Hagan and Peter Lawson Jones, and County Treasurer Jim Rokakis, previously have asked the pair to step down.
Both have denied wrongdoing and refused.
It's believed indictments of Dimora and Russo in the federal corruption probe may be coming soon.
Some Democrats wondered why Garson decided to make a request which is bound to draw more media attention to the party's high profile officeholders linked to the corruption probe.
Others speculated that he thought it best to put distance between the party and Russo and Dimora, who was its former chairman, as voters begin to make early voting primary choices for new county offices.
The controversy has spilled into the Senate race, where Republican Rob Portman is asking where Democrat Lee Fisher stands on Dimora and Russo's continuing in office.