The goal earned Crewe only their second away win as Orient slumped to their first home defeat in seven - but the home side only had themselves to blame as they failed to cash in on incessant second-half pressure.
It was Crewe who dominated the first-half and could easily have been more than one goal to the good at the interval.
After just four minutes Gary Roberts, on loan from Ipswich, fired in a free kick that Orient keeper Stuart Nelson parried and Maynard knocked the rebound wide.
Orient went close to levelling when top scorer Adam Boyd's header was deflected wide.
Crewe attacked with pace and poise and Maynard raced clear but shot wide before Byron Moore swept in a cross by Roberts only to have the effort ruled out for offside.
Orient lost defender Tamika Mkandawire through injury and substitute Sam Oji had only been on for 60 seconds when he failed to clear and Maynard reacted first to sweep the ball home.
Orient had a penalty appeal turned down when Corden's shot appeared to be handled.
As the O's stepped up a gear Boyd missed a glorious chance to level after being set up by Jabo Ibehre before Ben Williams made a wonder save on the stroke of half-time to keep out Ibehre's header.
After the break the O's took a vice-like grip and created chance after chance but goalkeeper Williams was heroic as he made stunning saves from Boyd, Ibehre and substitute Wayne Gray.
When he was finally beaten by Oji's header, the woodwork came to his rescue.
Boyd, Corden, Jason Demetriou and Paul Terry also wasted good chances for the O's, allowing Alex to walk away with all three points.