The Arnhem outfit took the lead inside seven minutes at the IJsseldelta Stadion when Chelsea loanee Lewis Baker slotted past keeper Micky van der Hart, but PEC levelled after quarter of an hour through Thomas Lam.
Lam's foul presented Vitesse with a penalty midway through the first half, which another Chelsea youth product, Dominic Solanke, converted to restore the visitors' advantage.
With the interval approaching, Georgian forward Valeri Kazaishvili netted a third as Vitas looked to be cruising to victory, with plenty to spare.
Zwolle's hopes of a recovery were dashed when Vitesse were awarded a second penalty for Ouasim Bouy's rash challenge, which earned the midfielder a straight red card.
Denys Oliynyk duly stepped up and sent Van der Hart the wrong way to all but seal victory for the visitors.
With 11 minutes left, Milot Rashica added a fifth for Vitesse, who leapfrog Zwolle in the table thanks to the convincing win.