Winnall struck from a pair of Gary McSheffrey free-kicks and his brace extended the Iron's unbeaten run under record-equalling boss Russ Wilcox to 23 matches.
Robins midfielder Matt Richards forced Sam Slocombe into his only save of note in the 36th minute with a header towards the bottom-right corner.
The opener arrived in the final minute of the first half. McSheffrey's free-kick was not cleared by the hosts and with Niall Canavan causing havoc in the penalty area, the ball fell for Winnall to hook past goalkeeper Scott Brown.
Winnall's second came just before the hour mark after Cheltenham failed to clear another dead-ball delivery from McSheffrey.
The home side missed a chance to force their way back into the game when Jermaine McGlashan was tripped by David Mirfin in the area shortly after Winnall's second. But Slocombe dived to his right to keep out Richards' spot-kick with a fine save.
Paddy Madden saw a fierce drive beaten away by Brown in the 81st minute but the Iron were able to coast home anyway.