The Premier League title holders lost 1-0 against Moyes' former club Everton on Wednesday and were beaten again on Saturday by a Newcastle side who picked up their first away league win at Old Trafford since 1972.
Yohan Cabaye was the man on target for Newcastle, with the France international scoring after an hour.
He calmly finished inside the area after being played in well by Moussa Sissoko to inflict consecutive home league defeats on the hosts for the first time since 2002.
Alan Pardew's side had a slice of luck, however, as Vurnon Anita looked to handball an effort off the line at 0-0, before Robin van Persie saw what would have been an equalising goal ruled out for offside.
Moyes made one change in defence, bringing in Jonny Evans to replace Chris Smalling while making wholesale changes in his midfield.
Nani, Phil Jones, Tom Cleverley and Adnan Januzaj were all brought in following the midweek loss, with Van Persie returning from a groin injury up front in the absence of the suspended Wayne Rooney.
Pardew made one change from his side's 3-0 reverse at Swansea City, bringing Anita into the midfield, with striker Shola Ameobi dropping to the bench.
Jones had the first effort of the game after eight minutes, forcing a comfortable save from Tim Krul, before Cleverley too had a shot inside the area that was well charged down by the visiting defence.
Loic Remy flashed a header wide for Newcastle, while Cabaye should have done better with an effort he dragged wide six minutes before the half-hour mark.
Pardew's side did well to stifle their opponents, limiting the defending champions to half-chances and, with Mathieu Debuchy seeing two efforts well saved late in the half, the scores were goalless at the break.
Krul was called into action in the second half, with Javier Hernandez getting a shot away after a lofted ball from Van Persie, before Patrice Evra's header was cleared off the line by Anita.
The Dutchman will consider himself fortunate, with the clearance appearing to come off his arm.
Things went from bad to worse for Moyes' men on the hour mark as Cabaye stroked home side-footed from inside the area to hand the visitors the lead.
Moyes' men applied the pressure late on but Newcastle held on to bounce back from their midweek defeat at Swansea in superb fashion.
