The Smoggies, beaten 1-0 by Aston Villa in the Welshman's first game in charge, let slip a one-goal lead but then recovered to take all three points back to the North-East.
There was further disappointment for automatic promotion chasers Cardiff City, meanwhile, as they lost for the fourth match in a row during the festive period.
Derby County could only slightly capitalise, though, with a 1-1 home draw to Sheffield United extending their unbeaten run to a seventh game, while fifth-placed Leeds United were held to a goalless draw by managerless Nottingham Forest.
Having recently put an end to their year-long winless home run, there was further disappointment for Sunderland as they lost 1-0 at home to Barnsley, while Hull City are now in even more trouble after going down 1-0 to Bolton Wanderers in a relegation six-pointer.
Championship results in full: Bolton 1-0 Hull, Derby 1-1 Sheffield United, Leeds 0-0 Forest, Norwich 2-1 Millwall, Preston 2-3 Middlesbrough, QPR 2-1 Cardiff, Sheff Weds 0-3 Burton, Sunderland 0-1 Barnsley