But Louis Briscoe equalised when Harris was unable to clear a cross and then Tyrone Thompson put Mansfield ahead when a blocked shot fell kindly.
Wrexham replied when Andy Morell headed in Harris' looped ball from close range before Harris, who had just struck the bar, blasted home a 25-yard winner.
The victory was the Dragons' fifth in a row, equalling their best run under manager Dean Saunders.
Source: BBC Sport
Source: BBC Sport