But Moreno should have not been on the pitch to head PSV level after Memphis Depay's opener.
With 15 minutes gone, the Mexican slid in with a late two-footed challenge on Shaw in the box.
The England defender was carried off on a stretcher while receiving oxygen eight minutes later. Disturbing replays showed Shaw's right foot bent at a 45-degree angle, which meant he had surely broken it.
Incredibly, referee Nicola Rizzoli did not award a penalty and Moreno was allowed to equalise Depay's stunning solo opener with a header.
Then, just before the hour, Narsingh took advantage of some slack marking from Shaw's replacement Marcos Rojo to finish off a counter and give the hosts an historic win that gives them an early advantage in Group B.
Even though Anthony Martial impressed on his full debut, manager Louis van Gaal will be disappointed that United's return to Europe's biggest club competition ended in a loss.
As a former Ajax coach and player, Van Gaal would have expected a hostile reception at the Philips Stadion.
Before the first whistle, it was clear the United manager and his team would not be made welcome by the supporters of Ajax's great rivals.
Van Gaal's name and that of former Ajax player Daley Blind were booed loudly when they were read out on the public address system.
The atmosphere turned flat in the 15th minute, however, when Shaw was sent flying into the air and dumped on to the turf by Moreno's challenge.
The left-back ran into the box at full pace only to be chopped down by Moreno from behind. He lay in agony on the floor with his right foot at an angle, which indicated a break.
After lengthy treatment, Shaw was carried off on a stretcher and driven to hospital.
With such a subdued atmosphere, the game struggled to spark into life.
Martial headed a free-kick wide, but, despite lots of possession, United lacked a killer touch.
Thankfully for Van Gaal, Depay was in the mood to show his former employers what they were missing. The Dutchman received a pass from Blind, skipped past Santiago Arias and Jeffrey Bruma before slotting the ball through the legs of PSV goalkeeper Jeroen Zoet.
As he had promised, Depay celebrated, but not wildly.
Any celebrations were short-lived, however, as PSV equalised five minutes later.
Moreno shrugged off Matteo Darmian to head in Maxime Lestienne's corner. Blind tried to clear but instead ended up helping the ball in with his head.
The game turned full circle as PSV took the lead after half-time. Lestienne was the provider again. He latched on to a crisp volleyed pass from Andres Guardado and floated a ball over Rojo's head for Narsingh, who nodded in.
United had a chance to reply soon after when Bastian Schweinsteiger chipped into the box, but Juan Mata mis-controlled with the goal gaping.
Luuk de Jong missed a good chance to score a third, but the PSV fans were happy to settle for a 2-1 win.
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