Not in US courts. It would be more accurate to say the UDHR overrides nothing and is not US law. It is a nonbinding UN General Assembly resolution; while it is very powerful persuasive authority and much is customary international law, it is persuasive authority only. See Sosa v. Alvarez-Machain, 542 U.S. 692.
The US has ratified the International Convention on Civil and Political Rights, which unlike the UDHR is a treaty, meaning that is a legal obligation for the US. However, the ratification was subject to many reservations and declarations, whose net effect is that the ICCPR is not in and of itself enforceable in US courts. The US considers the US Constitution to provide the rights in question, and basically assumes no further obligations.
Even if it was possible to enforce the ICCPR in US court directly, Reid v. Covert established that the Constitution overrides treaties in US court. While as a matter of international law treaties override domestic law, this is not necessarily enforceable in domestic court.