Drew Talbot's goal gave Chesterfield their sixth win in a row. And it moved John Sheridan's men to within a point of League Two's automatic promotion places.
The Spireites boss said: "We stopped them playing and did a very professional job. They are a good little team.
"I saw them play Burton the other day and was impressed, so this was a decent result. Overall we deserved to win."
Alex boss Dario Gradi said: "They got behind the ball and defended well. But we were poor in the second half and looked a bit tired."
-The Sun
Dario Gradi admitted he could do with a replacement for Calvin Zola as Crewe's strikeforce turned out to be a damp squib in last night's 1-0 home defeat by Chesterfield.
Zola, who has scored 16 goals this season, is sidelined with a hamstring injury - and his absence was keenly felt as Crewe offered little in attack.
Byron Moore hit the crossbar in the first half, but the Alex didn't threaten a well-organised defence again until left-back Matt Tootle was denied by Spireites keeper Tommy Lee in stoppage time.
- The Sentinel