Moore was cut to make room for tight Blake Jarwin, who was added to the active roster from the practice squad.
Jarwin's addition gives the Cowboys four tight ends on the roster, including Jason Witten, James Hanna and Geoff Swaim.
The Cowboys promoted Jarwin because the Philadelphia Eagles were trying to sign him and he is a player they didn't want to lose, according to a source.
The Cowboys might not be saying goodbye to Moore entirely. He is expected to signed back to the practice squad if he doesn't get picked up by another team.
Moore is still eligible for the practice squad.
The Cowboys play at Washington Sunday.