It's part of a plan county executive Ed FitzGerald unveiled earlier this year at his State of the County speech.
Councilman Mike Gallagher said he hoped to see the county take over more jails in the future.
"This is the first in many -- hopefully -- to come consolidations with the county jail and local municipality jails," Gallagher said.
Euclid will pay the county $400,000 a year for five years, plus another $600,000, according to a city council agenda. The county is also working out a similar agreement with the city of Cleveland.