Jesse Lingard scored another goal to continue his impressive run of form but despite holding a half-time lead, United were unable to keep their advantage.
In the stadium where they won the World Club Cup in 2008, not even the introduction to great acclaim of Shinji Kagawa, who amazingly was named man of the match, could inspire United to victory, with Yoshihito Fujita netting the winner three minutes from time.
Having named a youthful starting line-up, manager Moyes must have been delighted at the manner in which they responded to a quite unexpected early setback.
Their hosts piled forward straight from the kick-off but when David de Gea produced an acrobatic save to deny veteran Brazilian striker Marquinhos, United thought they had been let off the hook.
Instead, they struggled to regroup and when the ball was played back into the box, Marquinhos, the 37-year-old, was on hand to turn it home with a first-time effort.
Not even half a minute had been played.
It must have been a substantial positive for Moyes that none of those young heads went down.
Indeed, two of them combined to haul United level after 18 minutes.
Lingard scored twice in Sydney on Saturday and is evidently in form.
The 20-year-old played a pass out wide to Wilfried Zaha, then continued his run as the young England international burst into the box.
Masakazu Tashiro stretched to cut out the cross but Lingard pounced on the loose ball to take himself one clear of Danny Welbeck as the tour's top scorer.
Tashiro was not having the best of evenings and it was his own goal that gave United their advantage as he tried to clear a vicious Adnan Januzaj free-kick.
Jonny Evans failed to convert decent chances either side of the interval, the second in particularly should have been a goal, only for the Northern Ireland international to turn Zaha's cut-back wide.
United had cause to regret that miss as, with the crowd distracted by national hero Kagawa's first appearance to warm-up, Marinos defender Fabio climbed highest at the far post to head home direct from a corner.
Welbeck spured a glorious opportunity to restore United's lead virtually immediately and the visitors were still level on the hour when Kagawa was introduced.
The Japan star only started training properly yesterday after an extended break following the Confederations Cup.
Clearly, though, some kind of appearance was expected and he slotted in the middle of three attacking players operating just behind Welbeck.
Every touch was greeted with a sense of excitement and Kagawa almost set up a winner for fellow substitute Ashley Young, only for the England international's shot to get deflected into the side-netting.
A draw would have been a fair result but after good work from Jin Hanato, Fujita had the final word.