Second half goals from Presnel Kimpembe and Kylian Mbappe gave the impressive French side a clear advantage ahead of the return in Paris on March 6 and Pogba's dismissal for a second yellow card in the 89th minute makes the task even harder.
United had never previously lost at home in European competition by a margin of more than one goal and history suggests their task in the second leg is extremely tough.
No team has ever progressed having lost the first leg of a Champions League knockout tie at home by two or more goals.
It was United's first defeat under caretaker manager Ole Gunnar Solskjaer since he took over from the sacked Jose Mourinho in December and the Norwegian accepted it had been a reality check after 10 wins in 11 games domestically.
United made a highly aggressive start to the game but too often their forwards chose the wrong option or failed to deliver the right pass, although Marcus Rashford and Pogba tested PSG's 41-year-old keeper Gianluigi Buffon with near-post shots.
Thomas Tuchel's side, who were without injured forwards Neymar and Edinson Cavani, gradually gained the ascendancy - with Marco Verratti and Marquinhos dominant in midfield.
Mbappe should have hit the target when he was slipped through by Julian Draxler in the 28th minute but fired wide.
United had to play the second half without forwards Jesse Lingard and Anthony Martial, who picked up muscle injuries, and their night got worse in the 53rd minute when Kimpembe ghosted in at the back post to volley home a corner from Angel Di Maria.
Former United winger Di Maria, booed from the outset by sections of the crowd, had the last laugh when he collected a fine pass from Marquinhos and set up Mbappe for the second goal.
The Argentine broke down the left and delivered a low cross which was expertly turned in, first time, by 20-year-old France World Cup winner Mbappe who showed lightning speed to get ahead of the defence and turn the ball in with his right foot.
A disappointing night for United got worse when midfielder Pogba was sent off for a second booking after a foul on Dani Alves, leaving PSG in pole position to reach the last eight.
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AS Roma's Nicolo Zaniolo celebrates scoring their second goal. (Reuters)
Teenager Nicolo Zaniolo scored a quick-fire double to earn AS Roma a 2-1 win against Porto in their Champions League last-16 first leg on Tuesday.
The 19-year-old continued his superb breakthrough season by finding the net twice in six second-half minutes to become the youngest Italian to score two goals in a Champions League match.
Porto substitute Adrian Lopez, however, quickly responded with what could prove to be a crucial away goal.
The result ended a five-match winning run in Europe for Porto, who will welcome Roma to the Estadio do Dragao for the second leg on March 6.
