Lauri Dalla Valle headed the game's solitary goal to keep Crewe in touch with the play-off places and halt Carlisle's mini revival.
Carlisle defender Livesey fluffed his headed backpass and goalkeeper Mark Gillespie blocked Bradden Inman's shot only to watch helplessly as Dalla Valle headed the rebound over him and into the empty net.
The visitors, who had won their two previous matches, had the first sight of goal, but winger Matty Robson dragged his shot from the edge of the penalty area woefully wide.
Crewe striker Mathias Pogba teed up Luke Murphy for Crewe's first real opening, but the skipper fired high and wide.
Pogba then cut inside and curled a shot over the crossbar, while Byron Moore's snapshot was the first to trouble visiting goalkeeper Gillespie.
Carlisle almost snatched the lead when striker Lee Miller dummied a cross into the path of Jon-Paul McGovern, who hit his first-time effort straight at Crewe goalkeeper Steve Phillips.
But the Alex finally made the breakthrough two minutes before half-time when Dalla Valle reacted quickly to head home a rebound.
Carlisle almost equalised after the break, but Phillips produced a superb one-handed stop to keep out Miller's goalbound header.
Crewe defender Matt Tootle then cleared off the goalline following a goalmouth scramble and visiting substitute Rory Loy ballooned the rebound over from eight yards out with the goal gaping.