Yohan Cabaye swept home the only goal, moments after Patrice Evra headed against the post for the champions.
Robin Van Persie had a goal correctly ruled out for offside, while Adnan Januzaj and Wilfried Zaha also went close.
However, Newcastle produced a professional away performance to earn all three points for Alan Pardew's side.
The first half lacked many clear-cut chances, with the best effort seeing David De Gea forced into a smart save by Mathieu Debuchy just before the break.
United did not give Moyes the early response he would have been looking for following the 1-0 defeat to Everton on Wednesday night, with the returning Van Persie off the pace.
The second half started brighter for the hosts, with Tim Krul forced into a double save from Adnan Januzaj and Javier Hernandez.
Patrice Evra nearly took the lead with his header coming off the post, although replays showed Vernon Anita intervened with his hand to keep the ball out.
However, United's bad luck benefited Newcastle who took the lead shortly after through Cabaye.
Good work from Moussa Sissoko on the right flank allowed Cabaye a free shot at goal, which deflected off Nemanja Vidic to nestle in the back of the net.
Moyes threw on Wilfried Zaha for Nani, with the former Crystal Palace winger curling an effort just wide.
And Van Persie had a header ruled out for offside, to compound another difficult day for the new manager.
After the midweek defeat by Everton, David Moyes's side now sit 12 points adrift of Premier League leaders Arsenal.
A victory for Arsenal on Sunday will make it 15 points gap between league leaders Arsenal and former champions, Manchester United