The announcement came on the eve of President Petro Poroshenko's visit to the city to mark the 71st anniversary of its liberation by Soviet troops from the Nazis.
Ukraine's SBU security service said 29 people had been arrested and a large cache of arms and explosives was seized.
They were taking orders from pro-Russian separatist leaders in Ukraine's rebel-controlled east and were planning the murder of a lawmaker and several pro-Western activists, SBU chief Valentin Nalivaichenko said.
"According to their plan, this would be the start of the seizure of territory to wrest Odessa and its surrounding region and create a new quasi-republic," he said.
Two days earlier, the SBU had announced the arrest of 10 other people, all Russians, who they accused of planning bombings in the run-up to Friday's event in Odessa marking the liberation of the city during World War II.
They were "planning acts of sabotage and bombings," the SBU spokesman said.
Although Odessa lies far from the fighting in Ukraine's east, dozens of unclaimed attacks here have stirred up an atmosphere of mistrust in the already divided Russian-speaking Black Sea port.
Founded in 1794 by Russian Empress Catherine II, the bustling city lies deep in government-controlled territory but some here fear that it could eventually be in Moscow's sights.
Russian President Vladimir Putin last year said Odessa was not historically part of Ukraine but of Novorossiya -- a tsarist-era term now used by pro-Russian rebels for territory in the east.
More than 6,000 people have been killed in fighting between the separatists rebels and the Ukraine authorities that began a year ago this week.