The home side dominated much of the first half as Jamille Matt, Gareth Evans and Ryan Crowther all enjoyed chances to open the scoring.
But Chesterfield grabbed the lead against the run of play when substitute Armand Gnanduillet headed home Conor Townsend's corner in the 44th minute.
And from the restart the visitors doubled their tally, Richards on the end of a long ball before firing a stunning 25-yard effort which was too good for Scott Davies - capping a stunning finish to the first half.
Richards then made it three-nil just after the hour mark, lashing home the rebound after Davies saved Sam Togwell's goalbound effort.
Andy Mangan's first goal since August gave the home side hope, a perfectly timed run to get on to David Ball's defence-splitting pass and a cool finish.
Had Mangan found the net again, first shooting wide from 18 yards before then looping the ball over the bar from close range, Fleetwood might have set up a grandstand finish but it was not to be.
As it was Chesterfield stood firm and are certain to finish above the home side, who are winless in five, but Paul Cook's side cannot now catch Bradford in the fourth and final play-off place.