An entertaining first half failed to produce a goal at Gresty Road.
Former County striker Anthony Elding hit a post early on as Crewe started brightly.
But the Hatters came more into the game as the half wore on and Oli Johnson was unlucky not to give his side the lead when he flicked Danny Pilkington's shot against the inside of the post.
Two goals from Alex striker Colin Daniel in the opening ten minutes after the break gave the home side a strong advantage.
And when Ben Rix netted soon after it left County with a real mountain to climb to keep their Cheshire Cup hopes alive.
The Hatters gave themselves a lifeline on 66 minutes when Danny Pilkington squared the ball to Greg Tansey who beat Stuart Tomlinson with a side-foot shot from 10 yards.
Declan Edwards pulled another back two minutes from time when he followed up to score after Tomlinson had parried Johnson's shot.
And Crewe needed a flying save from Tomlinson to turn over Lee Edwards cross-shot in stoppage time to book their passage into the semi-final,
To watch Crewe's first two goals click here.
County: Rigby; Havern, Rose, Taylor, Raynes (Wilkie 45), J Thompson, Tansey, Blizzard (Jackson 75), Johnson, D Pilkington, Mainwaring (Edwards 8)
Unused subs: Astbury, Ennis.