Huke was one of two players left unmarked in the six-yard area and he stooped to head in Danny Foster's right-wing 63rd minute cross, the Daggers' first effort on target in the match, for his first league goal.
Michael Boulding squandered three clear chances to level for the poor Stags, who were made to pay for wasting a glut of first-half chances.
Stags defender Alex John-Baptiste almost marked his return from a six-match injury lay-off with an early goal, glancing Stephen Dawson's cross narrowly wide when unmarked inside the area.
Then Dawson miskicked a perfect John McAliskey lay-off with the goal at his mercy as the home side piled on early pressure.
A Micky Boulding piledriver was tipped around the post by the diving Tony Roberts and McAliskey should have scored from close range with Dagenham reeling.
Boulding crashed another long-range shot over after a neat McAliskey knockdown and Lee Bell drilled a low shot inches wide of the post.
Dagenham, restricted to rare breakaways in the first half, might have snatched the lead against the run of play when Chris Moore raced clear. But his poor control eventually allowed teenage defender Chris Wood to snuff out the danger.
Mansfield's sixth corner of the match after just 29 minutes ended with Jake Buxton's low drive blocked in crowded goalmouth.
Ten minutes later Matt Hamshaw's cross was helped on by Boulding and the stretching McAliskey guided the ball narrowly wide.
The Stags came even closer just 60 seconds on from their ninth corner. Boulding's header was knocked off the line by Scott Griffiths and John Mullins' two fierce follow-ups were blocked.
The game was held up for six minutes when Wood had to be stretchered off six minutes into the second half after a brave block on Huke.
His replacement Nathan Arnold fired a low shot across goal as the Stags continued to dominate.
But the home side were made to pay for wasting the hatful of chances when Foster's 63rd minute cross found the unmarked Huke and he headed in from six yards, Dagenham's first on-target effort in the match.
Boulding three times went close to an equaliser, but Dagenham hung on for an historic win.