The 19-year-old striker bagged his seventh goal in five starts this season when he edged the hosts in front midway through the first half.
A goalkeeping mistake by Ben Williams allowed Kayode Odejayi to level before the break for John Ward's side.
But Crewe captain Neil Cox struck a decisive second and man of the match Luke Varney rounded off the victory with the third.
Maynard latched on to a Jon Otsemobor slide-rule pass and pulled the ball in off the far post for the opener which capped a period of mounting pressure from Crewe.
David Vaughan had flashed a volley just wide and Cheltenham's early enterprise had disappeared.
But the visitors had a great opportunity to equalise on 36 minutes when Grant McCann teed up Brian Wilson 15 yards out, but the former Stoke midfielder saw Williams produce a stunning block.
Within two minutes the keeper seemed rooted to the ground when Wilson's shot-cum-cross from the right arrowed in at the far corner via the faintest of touches off Odejayi's boot.
Michael O'Connor almost restored Crewe's advantage just before half-time but keeper Shane Higgs kept out his angled drive with his legs.
Cheltenham really fancied their chances in the early period of the second half.
Jeremy Gill was given too much room to crack a rising drive from the right which Williams did well to get a glove to, but Higgs also showed excellence to keep out Varney's right-footer from finding the far corner.
Varney was Crewe's outstanding attacker but he let down his athletic approach play with some weak finishing. He shot straight at Higgs when he was put through by Otsemobor.
And Crewe could have been caught on the counter in the 65th minute when their former loanee Steve Guinan volleyed a foot wide and then McCann drove on to the roof of the net.
But when Guinan lost possession, Crewe surged to the other end of the pitch and Varney crossed for Cox to rifle in the second 20 minutes from time.
Dario Gradi's men sealed victory when another mistake gifted them possession. This time McCann fell over with the ball 10 yards outside the box allowing Gary Roberts to set up Varney for his first of the season from eight yards out.