Washington were beaten 27-23 by the Dallas Cowboys on Sunday (local time) to drop them to 0-2 and Cousin's wide receivers appeared frustrated with the quarterback, who could not convert in the red zone.
"We have total faith that Kirk will get it done, and he will get it done," Gruden told reporters. "He has proven that he can be a successful quarterback in this league."
Cousins has thrown for 693 yards in the two regular season games, the third highest in the league, though many have been accumulated with the team trying to catch up.
He has tossed just one touchdown to three interceptions.
The 28-year-old took over as the starting quarterback last season and recorded 4166 passing yards and 29 touchdowns while leading the Redskins to the playoffs.
The performance earned him a one-year franchise tag worth $19.95 million and increased scrutiny, which Cousins has not shied away from.
"You take ownership, you step to the forefront and you lead and you challenge guys and you say what you need to do better," Cousins told a Washington radio station on Monday.
"I think together we can do that. I've faced challenges before, adversity before on the football field.
"We're going to find a way to move forward and play better. We're going to do it together."
