The only thing that MUST arrive before the deadline for 99% of the universities is the online application form. All universities assume that supplementary materials (such as transcripts, TOEFL + GRE scores and recommendation letters) will arrive later, sometimes even a month after the deadline. This is perfectly fine.

In general, universities have different methods for processing the applications. Some review them on a rolling basis, which means that they don't have a definitive deadline and they start reviewing whenever a complete application is received. They continue this process until all the positions are filled. In this case, it is better to send the application materials as soon as possible (such as Case Western Reserve University).

On the other hand, other universities wait until about 5-6 weeks after the formal deadline, and then they start reviewing the applications in different sessions. This is the case in most universities and therefore, you should try to complete your application until about a month after the published deadline at the latest. Other universities (such as Berkeley) give you a specific date until which the supplementary material must arrive as well. For example, for the Mech. Eng. program at Berkeley, the online application deadline is Dec. 5th, and the deadline for receiving other material is Jan. 5th. However, I should point out that I called them and they said that some groups have not started reviewing the applications yet (as of Jan. 6th) and therefore, there was still time to complete the application.

So the most important thing you should worry about is completing the online application form and submitting it