Zola struck in the 34th and 45th minutes, Billy Jones added the third from the spot a minute after the break and Byron Moore added the fourth in the last minute.
Crewe's opener came from a breakaway when a Grimsby corner was cleared to Joel Grant on the right and he picked out Zola in the box. The big striker held off his marker before looping a right-foot shot over keeper Tommy Forecast.
Crewe stretched their lead when Moore crossed from the left and the unmarked Zola headed home.
A minute into the second half they made the game safe from the spot. Nick Heggarty pushed Moore in the box and Billy Jones slammed the spot-kick with his left foot.
Highly-rated Mariners defender Ryan Bennett saw a header smack against the post late on before Moore slotted home a loose ball for Crewe's fourth.