Wittenberg was 4-down after nine holes but the 18-year-old clawed back with four straight wins starting at the par-4 10th. He got to 1-up with a birdie at the 15th and was in position to close out the match until a bogey at the 17th put him all square with Rubinson.
The duo kept plugging along but Rubinson finally gave way with a bogey at the second, the 20th hole of the match.
'I played against Adam in the Western Amateur a couple years ago and he beat me,' said Wittenberg. 'He's a great guy. Unfortunately, somebody's got to lose. Today, I just held on a little longer.'
Reigning U.S. Mid-Amateur champion George Zahringer was 1-down after the opening hole but squared the match at the third and never trailed for the remainder of his match.
The 50-year-old used a birdie at the par-4 17th to defeat J.J. Killeen, 2 and 1. Zahringer will face Wittenberg in the quarterfinal round.
Patrick Carter outlasted Michael McDermott, 2 and 1, while Bill Haas earned a 2-and-1 victory of his own over Zach Atkinson to make the quarterfinals for the second straight year.
'My goal today was to advance on to the next day,' said Haas. 'You've got to set mini-goals each day. But I'm not setting myself up for failure; I'm trying to just give myself a challenge.'
Carter will meet Lee Williams on Friday. Haas will have his hands full with David Oh in the quarterfinals.
Oh, who defeated Trip Kuehne in the second round, didn't have much trouble in the third round and cruised to a 6-and-5 victory over Matt Kohn.
The other quarterfinal match will consist of 44-year-old Jerry Courville and 19-year-old Nick Flanagan.
The U.S. Amateur is one of 13 national championships conducted annually by the United States Golf Association, with 10 of the events strictly for amateurs.