With Wayne Rooney back in the team, United did not create much before the Turkish side gave away two cheap penalties.
World record signing Pogba smashed in after Simon Kjaer fouled Juan Mata before Anthony Martial converted following Sener Ozbayrakli’s push.
Pogba guided a superb curling shot home from 20 yards with Jesse Lingard drilling home from similar distance to put the game out of sight after the break.
The second half was played out in an exhibition atmosphere and Robin van Persie’s late consolation was applauded heartily as he met a low cross to mark his return to his former side with a goal.
United are level with Feyenoord on six points, two clear of Fenerbahce at the halfway point in Group A.
Pogba’s swagger holds sway
Pogba’s performances since his summer move from Juventus have yet to reach the heights that would justify his £89m fee, but his confidence appears to be building as he finds his niche in midfield.
With the composed Michael Carrick beside him, he was able to dominate the central third with his athleticism and energy.
He proved impervious to goalkeeper Volkan Demirel’s attempts to put him off before his spot-kick and showed the same cool head as he guided his second into the top corner.
Harder tests than a flimsy Fenerbahce side will come – not least the visit to Chelsea in the league on Sunday – but Pogba looks increasingly ready to meet and pass them.
“Two days ago he was the worst player in the Premier League and 48 hours later he’s phenomenal,” said manager Jose Mourinho, referring to criticism aimed at Pogba after his performance in Monday’s goalless draw with Liverpool.
“He needs time. I’m happy with his performance because he gave us what we need.”
Same opponents, a very different Rooney
Rooney’s inclusion – after being named as a substitute for the previous four games – gave the 30-year-old a chance to show he is as much a part of United’s future as its past.
Fenerbahce had been the opponents at the start of his United career, when he scored the first three of his 246 goals for the club with a debut hat-trick.
However England’s top scorer is a very different player from the 18-year-old that lit up Old Trafford on that night in 2004, and his best moments were in the build-up rather than finish of United’s goals.
Often operating deep and wide, a clumsy attempt to control a pass from Mata saw one one opening evaporate in an instant, but he then robbed a Fenerbahce defender to help set up Pogba’s second and assisted Lingard’s strike with a cushioned lay-off.
His leadership and maturity was also evident as he appeared to delegate penalty-taking to Pogba and help the Frenchman ease the pressure of his price tag.
With Sir Bobby Charlton’s club-scoring record just three goals away, it was a gesture that came at a personal cost.
Van Persie welcomed back
Van Persie’s return also evoked fond memories for the Old Trafford faithful.
The Dutchman finished as the Premier League’s top-scorer the last time United won the title in 2013, but, like the team as a whole, his form faded in the following two seasons before his move to Istanbul.
His name was chanted as he walked out for kick-off, he was cheered as he came across to take Fenerbache’s corners and, after curling a long-range shot just wide, he turned in Emmanuel Emenike’s cross to prompt an ovation from his former fans.
What they said
Manchester United manager Jose Mourinho: “Paul Pogba needs time. I was in Italy, I know Italian football. To be in Italy for four or five years and come back I was not expecting it to be a click of the fingers for intensity. He needs time.
“They came with a good defensive plan and stopped us for the first period but then after the first two goals, both clear penalties, it opened the door for quality.
“With the fourth goal we stopped and they tried to have a reaction which they did. I am never happy to concede a goal. We lacked a bit of concentration but that is the nature of the players and lots of changes, it is acceptable.”