I'm really hoping/praying/lighting my Alejandro candles for this situation: during the performance, Vevo quietly uploads the video, and then BAM Gaga ends the performance by announcing that her new music video is now available to stream online (or will be after the show at least)
After performing on The Voice, the video should be uploaded immediately, do a small interview with the host and tell the audience the video is up on youtube. At the same time, have youtube advertise the video like the Dope performance