French World Cup winner Pogba, relishing his football again since the change of management, took just five minutes to strike, sliding the ball in at the back post after a wonderful piece of dribbling and a precise low cross from Marcus Rashford.
Pogba, who scored twice against Huddersfield Town on Boxing Day, then burst into the box to head home an Ander Herrera cross from deep to make it 2-0.
There was real class about United's third, in the 45th minute, as Anthony Martial delivered a low cross with the outside of his boot and Rashford guided it home off the outside of his outstretched foot.
Bournemouth pulled a goal back in stoppage-time at the end of the first half though when David Brooks skipped around the United defence and placed a cross on the head of Nathan Ake who nodded past David De Gea.
Romelu Lukaku made it 4-1, two minutes after coming on as a 70th minute substitute, latching on to a Pogba through ball and hooking the ball past the advancing Asmir Begovic.
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Manchester United's Romelu Lukaku scores their fourth goal. (Reuters)
Pogba was inches away from a hat-trick with a low drive from 20 yards out that struck the post.
The relaxed and positive mood was slightly undone by a wild lunge in the 79th minute from United defender Eric Bailly on Ryan Fraser, with the Ivorian rightly being shown a straight red card.
Manchester City bounced back from two consecutive defeats to beat Southampton 3-1 on Sunday and cut the gap on Premier League leaders Liverpool to seven points before the top two meet on Thursday.
Pep Guardiola's champions needed to regain their footing after stumbling against Crystal Palace and Leicester, and did so with strikes from David Silva, Sergio Aguero and James Ward-Prowse's own goal at St. Mary's.
Chelsea kept up their pursuit of the Premier League's top three clubs with a 1-0 win away to London rivals Crystal Palace, thanks to a strike by N'Golo Kante early in the second half.
Goals from Chris Wood and teenager Dwight McNeil gave struggling Burnley a much-needed 2-0 Premier League win over West Ham United at Turf Moor, ending a run of three straight defeats for Sean Dyche's team.
