The Frenchman, who is ranked 36th in the world, will now face Andreas Seppi in the last 16.
Del Potro won just six receiving points during the opening set, as Chardy claimed the first three games of the match before serving out for a one-set lead.
Chardy continued his dominance in the second, hitting 16 winners and breaking twice.
Del Potro claimed his first break of serve in the opening game of the third set, but Chardy broke back immediately and a tie-break was required once again.
The Argentinian bossed the breaker, winning it 7-3, and he then continued his positive momentum by claiming the fourth set with a solitary break of serve.
Del Potro appeared to be the fitter player during the final set, but he was unable to cope with Chardy's ultra-attacking style, which saw the Frenchman land a further 16 winners.