But Darren Ferguson's men were guilty of making hard work of shaking off a struggling Alex side spearheaded by Chuks Aneke.
The Arsenal loanee opened the scoring after 13 minutes in spectacular fashion when guiding a curling shot into the far corner of the net.
But the leakiest defence in League One was breached only two minutes later as Harry Davis pulled back Washington and top scorer Britt Assombalonga stroked in his 25th goal of the season from the penalty spot.
Nicky Ajose finished a slick counter-attack involving Grant McCann and Danny Swanson to complete the turnaround four minutes before the break, but Aneke hauled Crewe level after 58 minutes when stylishly slotting past Bobby Olejnik.
However, his good work was quickly undone.
Swanson rifled Peterborough ahead on the hour mark with his first league goal of the season before January capture Washington opened his account for the club after flicking the ball past a defender and firing home.