Colchester got off to the perfect start when Freddie Sears poked home Ian Henderson's cutback with three minutes gone.
Marcus Bean saw a header saved by Alex goalkeeper Alan Martin, but the visitors doubled their lead with half an hour gone when Henderson latched on to Magnus Okuonghae's long punt and chipped home a sublime finish.
Five minutes after the break Crewe got the breakthrough as Pogba squared for Lauri Dalle Valle to pull one back with a simple finish into an empty net.
Martin then kept the home side in it by tipping Bean's lob over the bar before Murphy levelled from the penalty spot in the 64th minute after Magnus Okuonghae handled Dalle Valle's goalbound header.
Ten minutes later the Alex completed the turnaround when Aneke rose to head in Moore's cross at the far post.