United have it all to do in the Champions League last 16 second leg in Paris in two weeks' time after the defeat on Tuesday night, and they will be without midfielder Paul Pogba after he was sent off for two bookable offences.
Jesse Lingard and Anthony Martial had to be removed from the action due to injuries, and Solskjaer's squad may well be stretched for the return fixture at the Parc des Princes.
After a goalless first half, PSG opened the scoring on 53 minutes when Presnel Kimpembe volleyed home from close range from an Angel Di Maria corner.
Former United man Di Maria also played a key role in the second as the visitors doubled their lead seven minutes later.
Marquinhos started the move by dispossessing Pogba, PSG then worked the ball to the Argentine playmaker on the wing and Di Maria's perfectly weighted pass found Kylian Mbappe, who slotted the ball home from close range.
David de Gea prevented a complete collapse as he denied Mbappe and then Juan Bernat, with the Spanish stopper producing an exceptional save to prevent the latter from adding a third.
PSG sat back and defended in numbers in the final 20 minutes and managed to restrict their hosts to half chances, the best of which fell to Ander Herrera but his ambitious curling effort from the edge of the area failed to trouble Gianluigi Buffon in the away goal.
United were dealt a further blow late on as Pogba was dismissed when he picked up his second booking of the evening, meaning will be suspended for the second leg.