Tuesday's statement from the military says the helicopter had entered Turkish airspace by mistake while on a mission to monitor for cross-border infiltration of rebels. It was downed as it was returning to Syria.
Syria says Turkey "hastily" shot down the helicopter, thus revealing Ankara's "real intentions" of ramping up tensions.
Damascus says the helicopter had travelled a "short distance" in Turkish airspace before turning back.
Turkey's government has said the helicopter ignored repeated warnings to leave. It crashed inside Syria after being hit by missiles fired from a Turkish fighter jet.