James Collins' first-half header proved enough to earn Shrewsbury Town promotion to League One.
Graham Turner's men were deserved victors as they restored Town to the third tier of English football for the first time in 15 years.
But Swindon's home win over Port Vale ended any slim hopes of the League Two title for Town, who will now go up in second place.
Collins struck the only goal on 38 minutes, nodding in Nicky Wroe's cross.
In Turner's second season of his second spell as Shrewsbury boss, the former Aston Villa, Wolves and Hereford manager's triumph comes 33 years after his first promotion with the Shropshire club, who he led up to the old Division Two in 1979.
It was also the 64-year-old's sixth successful promotion campaign, having also twice taken sides up with Wolves and Hereford United.
In front of a sell-out crowd, Shrewsbury's victory also ensured that they have gone unbeaten all season at the Greenhous Meadow, where they have not lost now since 1 March 2011.
Shrewsbury made much of the running from the first whistle, skipper Ian Sharps curling an early shot just wide from the edge of the box before Jermaine Grandison's header from a Matt Richards corner was cleared off the line by Brian Woodall.
Dagenham's only first-half chance came when Medi Elito's 20-yard drive drifted narrowly wide.
But the breakthrough Shrewsbury craved eventually came when Wroe's angled cross was headed home by top scorer Collins for his 16th goal of the season.
Home winger Mark Wright bent a shot just wide in the closing stages as Shrewsbury stretched their timely unbeaten late season run to nine matches.
At the other end, Shrewsbury registered their 19th clean sheet of the campaign, with keeper Chris Neal never seriously tested before the champagne corks were allowed to pop.
FROM THE DRESSING ROOM
Shrewsbury Town boss Graham Turner told BBC Radio Shropshire:
"It was a surprise when we went up in 1979. And the expectations were greater this season, but we've got there in the end.
"I don't know if the game lived up to expectations. I know one or two more goals going in would have helped us breathe a little easier, but results went our way elsewhere.
"We were sound and solid defensively, but the occasion and the result were more important.
"And I've got to pay tribute to the players. They've been fantastic all season. For all the hard work they've put in, they deserve everything they've got."