Washington had struck twice in each of his two previous career appearances against Town and he needed only 80 seconds to continue that goalscoring streak.
He opened his account for the season when supplying a classy finishing touch to a low Harry Anderson delivery from the right.
Higher-level Posh were under the cosh late in the opening half and survived a major scare when Matt Harrold headed a Luke Rooney cross against a post.
The hosts had to wait until four minutes into the second period to double their advantage and Anderson was again the creator.
He shrugged off two challenges and slid the ball into the path of Marcus Maddison who coolly slotted past keeper Freddie Woodman from the edge of the box.
Crawley defender Sonny Bradley was millimetres away from providing a third Posh goal as he hooked a clearance against his own post.
Summer signing Joe Gormley then missed a glorious opportunity to grab a first goal in England football when stooping to head wide from close range, but it was ultimately a comfortable success for Posh.