Crewe's in-form striker Clayton Donaldson put the result beyond doubt with his 28th goal of the season shortly afterwards.
Anthony Elding should have given the Hatters the perfect start but shot wide from close-range after Patrick Ada misjudged Danny Rowe's cross.
Crewe striker Shaun Miller hit the upright after latching onto Donaldson's pass and Byron Moore's shot on the rebound was deflected to safety. Miller then headed a superb Donaldson cross wide at the near post before he came out second best in a one-on-one with County keeper Matt Glennon.
Elding turned another shot over the bar but the Alex went in front when Lee Bell's 61st-minute free-kick was nodded into his own net by the unlucky Goodall.
Four minutes later Donaldson picked up Danny Shelley's pass and beat Goodall before finishing low into the far corner of the net.