The goal was against the run of play in a first-half in which a young, new-look Mansfield side, including eight home debutants, completely dominated.
But, despite creating a wealth of chances, they just couldn't find a way past excellent keeper Jamie Young and an inexperienced Diamonds side who managed to keep their heads and their goal intact.
Greg Pearson had to clear a Matthew Tipton effort off the Rushden goal line as early as the third minute in a big scramble.
Then Young thwarted Stephen Dawson, Tipton, Gus Uhlenbeek and Adam Birchall with saves before Allen popped up in first-half stoppage time.
A well-worked corner routine saw Allen time his run towards the near post and neatly tuck away David Bell's corner to leave Mansfield stunned to find themselves behind at the break.
The second-half was a more even affair as Mansfield dominated possession, but, in throwing men forward, were always vulnerable to Rushden on the break.
The Stags changed their 4-3-3 formation to 4-4-2 and used all three substitutes as injured player-manager Carlton Palmer tried everything he could to find a way through.
Rushden keeper Young was again in action to deny Burchill at the near post and also keep out Fraser McLachlan's 25-yard thunderbolt.
But he saved the best for last and in the dying stages produced his two best stops, his fingertips keeping out Adam Rundle's low finish, and then somehow getting his body in the way of a Simon Brown blast.
But the Diamonds did have their chances to seal the win, with Bell firing over from another well worked set piece routine, and home keeper Kevin Pressman had to get behind a low Daniel Chillingworth shot before superbly tipping a Phillip Gulliver header over his crossbar.