The gunman, identified as Sheldon Francis, 29, of Middletown, was found dead in a nearby wooded area hours after the incident, police said.
The shooting took place around 10 a.m. at the Delaware Veterans Memorial Cemetery in Bear.
State police, EMS, and several other agencies responded to the scene.
Police found two victims suffering from gunshot wounds.
Delaware State Police Senior Corporal Heather Pepper said the 85-year-old woman from Elkton, Maryland was pronounced dead at the scene. The 86-year-old man subsequently died at the hospital.
Manny Vivar, who lives in the neighborhood next to the cemetery, said he got a knock on his door Friday morning from the state police.
"They told us to stay in place or evacuate," said Vivar.
It turns out, immediately after the shooting, the suspect ran through the woods and came out near his home.
Just a few yards away, authorities said there was a gunfight between state police and the suspect.
"It was real rapid like 'pow, pow, pow, pow, pow,'" said Vivar.
It appeared the windshield of a police armored truck was shattered, but no officers were hurt.
According to Delaware State Police, the suspect was found dead in the wooded area where the exchange of gunfire took place.
At this point, police said the motive is unclear.
They're also still trying to figure out if the suspect and victims even knew each other.
During Gov. John Carney's bi-weekly COVID-19 update to the state, he took a moment to reflect on the shooting that was unfolding in real-time.
"Really sad news to think of that kind of event in really what is a sacred ground, the final resting place of the men and women who have served our country," said Delaware Governor John Carney.