Moore, who was selected with the 24th overall pick of the 2018 NFL Draft, lost the ball on a punt return and again on a catch that led to a field goal.
"Just have to be aware," the 21-year-old Moore said, "and hold onto the ball tighter."
Moore likely will hear that and more from Panthers head coach Ron Rivera over the following week.
"DJ is going to be a part of what we do for a long time, as is Curtis Samuel," Rivera said. "If we believe in them, we're going to keep putting them out there. That's the only way these guys are going to learn and develop into the players we believe they can be.
"We will stick with them."
Moore's first fumble came on a punt return after Washington failed to record a first down on its opening series. He fielded the punt at his own 21-yard line, but Jeremy Sprinkle forced it out and made the recovery.
"Just too much effort on the punt [return]," Rivera said. "He should have protected the ball and gone down."
The second fumble occurred early in the second quarter at the end of a 17-yard catch and run. Former Panthers cornerback Josh Norman punched the ball out of Moore's grasp and linebacker Mason Foster recovered at the Washington 33-yard line.
"Punched the ball out. Something Josh Norman learned from Charles 'Peanut' Tillman," Rivera said of the former Carolina cornerback. "Got to give credit for a play well made."
Moore earned Big Ten Receiver of the Year honors in 2017 after setting a single-season school record at Maryland for receptions with 80. He posted 1,033 receiving yards and eight touchdowns, becoming just the third wide receiver in school history to have a 1,000-yard receiving season.
The 6-foot, 210-pound Moore started 35 of the 37 games he played and finished fifth in school history in career receptions (146), sixth in receiving yards (2,027) and fourth in receiving touchdowns (17).