Wasps winger Wade, who has one senior cap, dived over in the corner five minutes from time at Outeniqua Park after a driving maul took the ball within a few metres of the tryline.
Flyhalf Danny Cipriani, another Saxon who has been capped by England, converted to complete a stirring comeback after the tourists trailed 26-8 six minutes into the second half.
The Saxons, who won 32-24 in Bloemfontein last weekend, responded to the 18-point deficit with tries by hooker Tommy Taylor, flanker Matthew Kvesic and Wade, and Cipriani converted them all.
