Byron Moore handed the Railwaymen a second-half lead but Dagnall, signed from Barnsley just 48 hours before, hit back with two goals inside seven minutes.
Crewe shaded the first half and forced loan keeper Eldin Jakupovic into two smart stops.
The Serbian did well to hold a rasping drive from Brad Inman in the first minute and he clutched onto Ajay Leitch-Smith's snapshot.
Orient were struggling to live up to their lofty billing and Moses Odubajo failed to trouble keeper Ben Garratt when he was teed up by Dean Cox.
And the high-flyers were almost punished when Chuks Aneke brought down a high ball and skipped inside his man only to blaze over with Jakupovic's goal gaping.
Stoke loanee Jamie Ness almost opened Orient's account at the restart but the Scot's fierce drive was diverted off the line by some last-ditch defending.
And the Londoners fell behind after Aneke slipped behind the last man and when his powerful shot was palmed up in the air by Jakupovic, Moore was running in to apply a close-range finish at the far post in the 52nd-minute.
But when Cox slipped Dagnall clear on the left of the box the former Barnsley man found the far corner with a clinical finish on the hour.
And he was in the right place at the right time to volley Kevin Lisbie's lay-off beyond Ben Garratt seven minutes later.