It was announced on Monday that Mislintat will leave next month having only been at Arsenal for a little over a year but Emery believes the Gunners should take their time in appointing a successor.
Roma's director of football Monchi has been linked to a similar role in north London, but Emery says mistakes can be made if a process is rushed.
Asked if it was important to swiftly find a replacement for Mislintat, Emery replied: "Maybe if you do this recruitment very quickly, maybe you can make a mistake.
"I think it's better to do without that, being calm and also finding the best possible person to do this job like we want.
"I think the club is working on that. It's not my issue. I can say the same: if the club decided to sign one sporting director, then I will agree with them."