Mr Trump made the announcement after Tuesday's historic summit with North Korean leader Kim Jong-un in Singapore.
The move is seen as a big concession to North Korea and appeared to take US allies in the region by surprise.
Meanwhile, North Korean state media said Mr Kim had accepted an offer from President Trump to visit the US.
KCNA news agency said Mr Kim had invited Mr Trump to visit Pyongyang "at a convenient time" and Mr Trump had also invited Mr Kim to the United States.
"The two top leaders gladly accepted each other's invitation," KCNA added.
In his first reported comments since the talks, Mr Kim said it was "urgent" to halt "irritating and hostile military actions against each other".
He said the two countries "should commit themselves to refraining from antagonising" each other "and take legal and institutional steps to guarantee it", KCNA reported.
The Kim-Trump summit - the first such meeting between the two countries - centred on nuclear disarmament and reducing regional tensions. It concluded with a one-page agreement.
In the press conference afterwards, however, Mr Trump added another announcement: the cancellation of the military exercises.
The drills, often called "war games", are held in South Korea with local forces and US soldiers stationed there.
Mr Trump also said he wanted to bring US troops home - although he did not specify a timeframe.
He said the drills were "provocative" - even though the US has previously defended them - and there were suggestions that US allies had not been forewarned of this move.
South Korea's Presidential Blue House said it needed "to find out the precise meaning or intentions" of Mr Trump's statement.
South Korean President Moon Jae-in spoke by phone with Mr Trump for 20 minutes late on Tuesday, but an official readout of the call made no mention of military exercises, according to Reuters.