Pogba played for the full 90 minutes after his return to United in a world-record 89 million pounds (105 million euros, $116 million) transfer from Juventus, but it was Ibrahimovic who stole the show.
The Swedish star scored with a first-half header and second-half penalty to make it four goals in three games for Jose Mourinho's side, who took a provisional three-point lead at the top of the table.
"We were good enough to deserve the victory and be in control. We won comfortably and the result could have been bigger," Mourinho told Sky Sports.
"Zlatan Ibrahimovic is very good in the air. But he's great with his feet. I'm really happy with his performance.
"Paul Pogba was so good. He's learning how to play with us. I couldn't be happier with him."
United were deserved winners and with Eric Bailly and Henrikh Mkhitaryan also making their competitive home debuts, there were signs that the page is being turned on the drab days of the Louis van Gaal era.
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Southampton had won 1-0 on their previous two visits to Old Trafford, but they did not seriously threaten to make it three in a row in what was the first of 10 Friday games in the Premier League this season.
Having seen his side draw 1-1 with Watford last weekend, new Southampton manager Claude Puel is still waiting for a first win.
"We finish this game with a lot of regrets. We played good football with good opportunities," said Puel.
"There were lots of things that were positive for the future."
Pogba, making his first United appearance since a 5-0 win at Wolverhampton Wanderers in March 2012, lined up alongside Marouane Fellaini at the base of midfield in a 4-2-3-1 formation.
His name drew huge cheers from the home support when it was read out prior to kick-off, but his first touch was forgettable -- an ungainly air-kick that enabled Southampton to spring a counter-attack.
Things quickly improved, however. He registered United's first shot on target and then found Juan Mata with a clever, chipped pass, the Spaniard flicking the ball on for Ibrahimovic to volley acrobatically wide.
Making a statement: Perfect start for Mourinho as United win 3-1
Southampton suffered an early blow when midfielder Oriol Romeu was forced off by injury, Jordy Clasie taking his place, but their pace on the break caused the home side some anxious moments.
Shane Long threatened twice, first ruffling the side-netting and then scuffing a shot straight at David de Gea after Mata's under-hit pass had enabled Dusan Tadic to send the Irishman scampering clear.
Mourinho's pre-match call for United's fans to create an intimidating atmosphere was not observed to the letter, but in the 36th minute the players generated some noise on their own.
Antonio Valencia prodded the ball into space on the right, Wayne Rooney whipped a cross to the back post and Ibrahimovic leapt above reported United target Jose Fonte to plant a towering header past Fraser Forster.
Tadic gave United a scare by beating De Gea with a header early in the second half, but an offside flag spared the hosts.
After Anthony Martial had warmed Forster's gloves at the other end, Ibrahimovic made it 2-0, sending the Southampton goalkeeper the wrong way from the spot after Clasie tripped Luke Shaw.
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Pogba had a number of chances to extend United's lead, first heading over from Mata's corner and then blazing a pair of shots off-target.
Martial, meanwhile, took too long sizing up a chance created by Rooney, allowing Virgil van Dijk to block.
Mourinho handed a competitive home debut to Mkhitaryan late on, before Chris Smalling made his return from suspension as a replacement for Rooney.
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