Carlisle's automatic promotion hopes were kept alive thanks to Joe Garner's second-half strike. The deadlock was broken when full-back Zigor Aranalde crossed from the left wing and Garner tapped in from six-yards past Ben Williams.
The home side dominated and only goalkeeper Williams' heroics prevented Simon Hackney from squeezing a shot inside the upright.
Crewe's best chance came late through Neil Cox's dipping 25-yard free-kick.
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