Wrexham started brightly with Chris Llewellyn putting them ahead early on, but they struggled through the first half and the advantage always seemed in danger.
Llewellyn had an early shot wide after a Matt Done corner and defender Steve Evans tried a long shot that flew too high.
At the other end in a weak start, Rochdale could only produce a header from Keith Barker but it was a sitter that he should have put away. Instead he gave Michael Ingham an easy save.
Wrexham tried to push ahead and from a Lee Roche cross, John McAliskey flicked on and Llewellyn shot wide of the left post.
Again when Matt Crowell aimed a free-kick into the box, Llewellyn pulled it back for Mark Jones who fired low and wide.
Wrexham hoped for an improvement in the second half, but within three minutes Rochdale were level.
The goal had been on the cards for sometime and it was no surprise when Gary Jones fired a firm pass through for Dagnall to finish at close range.
Rochdale began to look extremely dangerous and Morike Sako might have done better when he led a breakaway following a Wrexham corner and chose to shoot when team-mates were better placed and the lively Dagnall expected a cross that would have put him clear on goal.
Dagnall added his second goal from a follow-up when goalkeeper Ingham spilled a shot from Alan Goodall at the foot of the post.
Wrexham were struggling in midfield, although their defence looked solid until faced with extra pressure brought about by the lack of cover in front.
Veteran Darren Ferguson came close to pulling a goal back with a low shot that Matthew Gilks went down to save as the ball dropped and skidded on the turf.
Wrexham switched to match Rochdale's formation, pushing three upfront with Josh Johnson replacing Done for the last half hour but it made little difference.
The normally lethal Mark Jones tried a solo run and shot only to see the ball curl wide of the far post and, when Ferguson crossed for Evans to head back across the posts, Llewellyn could only head wide.