It says the drone was on a routine mission in international airspace when two Russian jets tried to intercept it.
The US European Command said Tuesday's crash was the result of an "unprofessional act by the Russians".
Russia said the drone crashed after a "sharp manoeuvre", and denied that the two aircraft made direct contact.
The Russian defence ministry also said the MQ-9 Reaper drone was flying with its transponders turned off. Transponders are communications devices that allow the aircraft to be tracked.
Reaper drones are surveillance aircraft with a 20m (66ft) wingspan.
The incident happened at about 07:03 Central European Time (06:03 GMT) on Tuesday, according to the US military.
"Our MQ-9 aircraft was conducting routine operations in international airspace when it was intercepted and hit by a Russian aircraft, resulting in a crash and complete loss of the MQ-9," the statement said.
Several times before the collision the Su-27 fighter jets dumped fuel on the drone in a "reckless, environmentally unsound and unprofessional manner", it said.
The US has summoned the Russian ambassador in Washington to protest against the move.