The young striker completed his first senior hat-trick in his 50th game for the Railwaymen, taking his tally for the season to 15 goals.
But Crewe's progress wasn't secured until Maynard struck twice in the space of eight second-half minutes after Steven Gillespie had raised Cheltenham's hopes with a fine chipped finish.
Crewe were the better side in the first half and Maynard got them underway with a super finish from the left side of the box in the 14th minute after he had been found by a searching ball from Kenny Lunt.
Cheltenham were fortunate to escape on the half-hour when centre-half Shane Duff clearly tripped Tom Pope as the Crewe striker tried to reach a pass from Maynard on the right of the box.
And the visitors almost fell two behind just before the interval only for Ashley Vincent to hack Julien Baudet's shot on the turn off the line.
Keith Downing's side were vastly improved after the break and applied persistent pressure to the home box before Gillespie was set up by Vincent. After throwing a feint to undo a couple of Crewe defenders he clipped the ball beyond Ben Williams into the far corner.
But Crewe hit back within six minutes when Steven Schumacher and Pope tried to work their way in on goal, Maynard shielded the ball away from Duff and crashed a low shot into the bottom corner.
The 20-year-old completed his hat-trick in the 76th minute when he reacted quickly on the goal line after Baudet's crashing header from Lunt's corner could only be parried by Shane Higgs.
Gillespie scuffed a late chance to reduce the arrears and was then denied by a brilliant reflex save from keeper Williams, but it was Crewe's day and the victory means they are now within a point of guaranteeing League One football next season.