Steve Davis had called on his men to end a run of three successive draws and they did just that with two first-half goals as the Sky Blues, who could have ensured safety themselves, meekly surrendered to set up a nerve-jangling last day.
The hosts had started by far the stronger of the two teams and carved out four clear chances inside the first 15 minutes but never looked capable of coming back once they had conceded.
Lauri Dalla Valle put the visitors in front in the 23rd minute when home goalkeeper Lee Burge failed to keep out a near-post cross, allowing the Finn to tap home.
Burge should have done better 12 minutes later too when he dropped a cross which allowed Ryan Colclough to fire home.
Coventry made three early changes after half-time, but they showed little in attack and surrendered again in the 73rd minute as a fine Crewe move saw Marcus Haber head home from 15 yards out.
Tony Mowbray’s side managed a consolation five minutes from time when Nick Proschwitz headed home JIm O’Brien cross.