Russia said the attack occurred at about 4am and that its air defences had successfully shot down the unmanned aircraft.
Debris fell on the Expo Centre, Moscow mayor Sergei Sobyanin said on his official channel on Telegram.
The Expo Centre is a large space used for massive exhibitions, less than 5km away from the Kremlin.
A Reuters journalist in the area said he heard a "powerful explosion" in the business district area near the centre.
A video published by Russian media outlets showed thick smoke rising next to skyscrapers.
The Russian defence ministry said that Ukraine was behind the drone attack.
Russia has launched thousands of drones at Ukrainian cities, energy infrastructure and military sites since its February 22 invasion. However, Ukraine has, in recent months, developed its own long-range attack drones for retaliation.
A series of drone attacks in Moscow at the end of May damaged several buildings in the city.
Russia relies on drones – often Iran-designed and manufactured Shahed 129 aircraft, for cheaper, long-range attacks – to supplement its cruise missile arsenal.
Missiles are far more costly and experts say Russia is struggling to manufacture them in significant quantities, subsequently forging stronger ties to Iranian drone producers.
In response, Ukraine has ramped up development of its own long-range attack drones.
Moscow is about 450km from the Ukrainian border, well within range of new Ukrainian drones such as the UJ-22, which has a range of 800km.
Drones are extremely difficult to shoot down because they often fly at very low altitude under radar beam cover, in addition to being quiet and presenting very small targets for air defence.