France midfielder Pogba scored a penalty and curled in from distance to take his tally to three goals since his world-record 89 million pounds (USD 109.1 million, 100 million euros) return from Juventus.
Anthony Martial, with another penalty, and Jesse Lingard were also on target as Jose Mourinho's team secured a win that left them level on six points with Feyenoord at the top of Group A.
Fenerbahce replied late on through United old boy Robin van Persie, who tucked in a cross from substitute Emanuel Emenike and was rewarded with applause from all four corners of Old Trafford.
The result vindicated Mourinho's decision to make seven changes to the team that had held in-form Liverpool to a 0-0 draw on Monday.
The likes of Zlatan Ibrahimovic, Ander Herrera and Marcus Rashford are likely to return for Sunday's Premier League trip to Chelsea, after which United host Manchester City in the League Cup.
The joint-biggest win of the nascent Mourinho era was witnessed by ex-England manager Sam Allardyce, who attended as a guest of Alex Ferguson in his first public appearance since his much-publicised departure.
Dropped for United's last three games, captain Wayne Rooney returned up front to face the team against whom he scored a memorable debut hat-trick in a 6-2 Champions League win in September 2004.
With Juan Mata prominent in the number 10 role, United's build-up play was neat, but they showed next to no threat until they were awarded a 31st-minute penalty.
Michael Carrick's raking pass was beautifully brought down by Mata, who was brought down in turn by a clumsy challenge from the stumbling Simon Kjaer.
Referee Benoit Bastien thrust his hand towards the penalty spot and after an almost comical tip-toed run-up, Pogba tucked away a low shot to goalkeeper Volkan Demirel's right.
The home fans had called for Rooney to take the spot-kick and when Bastien awarded United a second penalty three minutes later, they were disappointed again.
Mata's exquisite, fizzing pass freed Martial, who was caught from behind by Sener Ozbayrakli, and the Frenchman picked himself up to deliver the sentence in person.