In a video interview on Ukrainian television, the ten Russian paratroopers said they had been on patrol of the border in Russia when they accidentally crossed into Ukraine near Dzerkaine, about 50 km (30 miles) southeast of rebel-held Donetsk.
"I did not see where we crossed the border. They just told us we were going on a 70km march over three days," one soldier, Ivan Milchakov, said in the video. "Everything is different here, not like they show it on television. We've come as cannon fodder."
On Monday Ukrainian military spokesman Andriy Lysenko accused Russia of trying to cover for the mistake of getting caught -- not the mistake of violating Ukraine's border.
"If elite troops do not know topography and do not know their locality, I can say nothing about that," he said. "The armed forces, the generals sent people to the east of Ukraine."
"We believe that was not a mistake, but a special mission they were carrying out," Lysenko said.
A Ukrainian television report that carried the interviews with the men said they were from the 331st regiment of the 98th Svirsk airborne division.
On Russian state television, a Russian defense source said the captured soldiers' story was true.
"The soldiers really did participate in a patrol of a section of the Russian-Ukrainian border, crossed it by accident on an unmarked section, and as far as we understand showed no resistance to the armed forces of Ukraine when they were detained," the source said.
The insider also claimed 500 Ukrainian forces had crossed the border the other way, but Russia had not announced it.
"We did not give much publicity to that," the source said. "We just returned all those willing to return to Ukrainian territory at safe places."
Ukraine also said Monday it was fighting with Russian forces who had crossed the border near Novoazovsk in an armored column.
Lysenko said 12 Russian armored infantry carriers had been destroyed, but 12 Ukrainian military personnel had been killed in the previous 24 hours, including four border guards he said were shot at by Russian Mi-24 helicopters.
The clashes come as Ukrainian President Petro Poroshenko and Russian President Vladimir Putin are due to meet for a summit in Minsk, Belarus.
Geoffrey R. Pyatt, the U.S. Ambassador to Ukraine, warned the multiple incidents appeared to be connected.
"The new columns of Russian tanks and armor crossing into Ukraine indicates a Russian-directed counteroffensive may be underway. #escalation," he tweeted.
Some 2,000 people have died in the fighting since early this year, when Russia claimed Crimea and separatists pushed to join them in eastern Ukraine. Ukraine has repeatedly accused Russia of arming the rebels, and even sending forces to fight with them, a charge Russia denies.
In the video of the captured soldiers, on Russian soldier, a Sgt. Andrei Generalov seems to corroborate Ukraine's claims, and expressed weariness with the months-long conflict.
"Stop sending in our boys," Generalov said. "Why? This is not our war. And if we weren't here, none of this would have happened."