Had Crewe lost, they would have been eight points behind Yeovil with three games remaining.
But after Maynard scored in the eighth minute, it turned into a torrid afternoon for the home side.
Maynard skipped passed Terrell Forbes on the left side of the box to curl a beauty in off the far post of goalkeeper Steve Mildenhall, who returned to the Yeovil side after injury when Portsmouth recalled loanee Asmir Begovic.
Crewe had to wait until the 57th minute for their second goal and again it came from the ever-dangerous Maynard.
Steve Schumacher cleared out of defence and Maynard got goal-side of Forbes in a two-man race before superbly lobbing over the advancing keeper.
Yeovil created precious little near goal and their only chance in the second half came on 74 minutes when Lloyd Owusu lifted a close-range effort against the underside of the bar following a free-kick.
Four minutes later, Owusu's contribution ended when he was stretchered off following a tackle and the Yeovil horror show was completed in stoppage time when the impressive Tom Pope got in the third with the defence stretched.
Yeovil manager Russell Slade said: "There were too many players out there today who didn't put a shift in, so when I get the opportunity they will be shifted out."