The Blades had already booked their play-off place but Aneke's winner means they have won just one of their last seven games.
Harry Maguire almost gave the visitors an early lead but his header was saved by Steve Phillips and Kelvin Mellor did well to deflect Neil Collins' follow-up over the bar.
Jonathan Forte sent a couple of efforts from the edge of the box high and wide before Phillips was in action again.
The keeper dived low to parry Callum McFadzean's drive and then recovered quickly to smother Chris Porter's rebound.
Crewe's Oliver Turton and Harry Davis sent efforts off target while Aneke's daisy-cutter was tipped around the post by George Long as the home side struggled to create any clear-cut first-half chances.
But Aneke stunned the visitors four minutes after the break when he beat Long with a superb 25-yard finish to put the Alex in the driving seat.
A fierce Barry Robson free-kick which Phillips did well to palm away almost got the Blades back in it while the keeper got a fingertip to a late shot from Joe Ironside.
At the other end, Aneke looked to double his account with a volley which dipped over the bar and a low effort which he dragged wide after a surging run.