James Collins headed the Shrews into League One as they completed a season unbeaten at the Greenhous Meadow.
Shrewsbury made much of the running from the first whistle and Ian Sharps curled an early shot just wide from the edge of the box. Graham Turner's men went closer still when Jermaine Grandison's header from a Matt Richards corner was cleared off the line by Brian Woodall.
But the breakthrough Shrewsbury craved eventually came in the 37th minute when Nicky Wroe's angled cross was headed home by top-scorer Collins for his 16th goal of the season.
Dagenham's only first-half chance came just earlier when Medi Elito's 20-yard drive drifted narrowly wide. Home winger Mark Wright bent a shot just wide in the closing stages, while at the other end Shrewsbury registered another clean sheet with goalkeeper Chris Neal never seriously tested.
Turner, in his second spell as Shrewsbury manager, has now enjoyed two promotions in charge of the Shropshire club after a gap of 33 years.