Substitute Zirkzee poked home an Alejandro Garnacho cross two minutes from the end of the game, on a night where Noussair Mazraoui and Matthijs de Ligt also made their Old Trafford bows.
Prior to the Dutchman's goals, United had the better of the opportunities, with Bernd Leno denying Bruno Fernandes twice in the first half, before stopping a Mason Mount effort just after break.
Andre Onana made a couple of acrobatic saves at the other end, stopping shots from Kenny Tete and Rodrigo Muniz, during an entertaining encounter that looked like ending goalless before Zirkzee's timely flourish.