MOSCOW—Russia said a marine was killed when one of its helicopters was fired upon during a search and rescue mission for the pilots of a Su-24 jet fighter downed by Turkish warplane earlier Tuesday.

“With the aim of evacuating the Russian pilots from where they landed, a search and rescue mission was carried out with the use of two Mi-8 helicopters,” Lt. Gen. Sergei Rudsky said during a Defense Ministry briefing.

“In...