Pogba had missed United's previous 10 games due to a thigh issue suffered in the 3-3 home draw with Everton in early February, yet he did not take long to make an impact on proceedings in the second leg.
Having replaced Marcus Rashford at the break, the France international cleverly fired in a loose ball in the 48th minute to secure his side a 2-1 aggregate triumph.
Zlatan Ibrahimovic - back from injury himself in time to feature off the bench – was unable to help Milan manage a response following his introduction, meaning the Serie A side have now failed to win five in a row at home.
United had triumphed 4-0 on Italian soil in the previous round, albeit that was against Spanish opposition in Real Sociedad, with their tie moved to Turin due to travel restrictions in place due to the coronavirus pandemic.
It started brightly in Milan too, yet it was the host which created chances towards the end of the first half. Alexis Saelemaekers tested Dean Henderson from distance, while Rade Krunic volleyed wide under pressure just before half-time.
The host – as it had done at Old Trafford – was assuming control for long periods.
However, Pogba punished Milan after they had failed to clear Daniel James' deflected shot, using a feint to create the space to lift the ball over goalkeeper Gianluigi Donnarumma from a tight angle.
Henderson came to United's rescue to deny Ibrahimovic, and Pogba came close to doubling both his and the visitors' tally, looping a header onto the roof of the net. In the end, though, one goal was enough.