Four grenades were dropped on the court and 15 shots fired at the interior ministry on Tuesday, officials said.
Rogue policeman Oscar Pérez said he had piloted the stolen helicopter to attack what he called a “criminal government”. His whereabouts are unknown.
Venezuela is in the midst of a political and economic crisis that has sparked mass protests.
A blue police helicopter was seen flying over central Caracas carrying a banner reading “350 Freedom” – a reference to a clause in the constitution cited by opponents of Mr Maduro to claim his government is illegitimate.
Images on social media showed two occupants, one masked.
The government said 15 shots were fired at a social event at the interior ministry. The helicopter then flew to the court and dropped four Israeli-made grenades of “Colombian origin”. One failed to detonate.
President Maduro appeared on state television to denounce the attack.
He said: “I have activated the entire armed forces to defend the peace. And you can be assured that sooner or later, we are going to capture that helicopter and those that carried out this terror attack against the institutions of the country.”
Information Minister Ernesto Villegas said the military had been deployed to apprehend the pilot and recover the German-built Bolkow helicopter.
State TV showed images of the pilot in front of the US Capitol building in Washington.
Mr Maduro has long claimed the US government is attempting to have him overthrown.
The government has urged calm but called for its supporters to rally outside the presidential palace on Wednesday morning.