The reigning champions found themselves 2-0 down within 15 minutes at the KC Stadium, with James Chester volleying home against his former club before David Meyler's strike found the net via a deflection from Jonny Evans.
United refused to wilt, however, and soon restored parity thanks to Chris Smalling's header and a superb volley from Wayne Rooney - his 150th goal in the Premier League for United - before Chester headed the decisive goal into his own net from Ashley Young's cross.
"To be two down and come back to win is a great result," Moyes told BBC Sport. "I thought it would be a long road back but these players have done this in the past and we did well to get back in the game.
"For us to get in at half-time 2-2 was great but when we were at 3-2 Hull gave us a few scary moments. However, I'd take that after being 2-0 down after 10 minutes.
"We're going to try to get the gap as narrow as we can. There's a long way to go in the league and we have to keep picking up points.
"There's a lot to do but we'll keep at it. I believe we can be in the mix and challenge whoever is near the top."
The victory was United's third win in a row in the Premier League, and they now have 17 Premier League wins from their last 18 Boxing Day fixtures.