However, visiting Posh responded positively to having fallen behind and in the 11th minute they were level. Crawley goalkeeper Jamie Ashdown made an initial save, but Michael Smith was on hand to fire in the rebound from close range.
Both sides then had opportunities to add the game's third goal, which was eventually scored five minutes before the break. Jon Taylor's cross evaded everyone, but not Peterborough defender Christian Burgess as he was on hand to put his side 2-1 up.
It was a goal that seemingly lifted the visitors, who went on to score two more times shortly after the restart. First Marcus Maddison profited from a mistake by Ashdown to score his fourth goal since joining Peterborough from Gateshead, before Conor Washington created himself enough space to slot in from 12 yards out in the 58th minute.
Those quickfire goals ended the match as a contest as Darren Ferguson's away side saw out the remaining minutes with relative ease.