United had the first opportunity of the match in the sixth minute. Luke Shaw floated the free-kick to Harry Maguire at the far post and United's captain headed it back towards the middle. It landed to Fred, who forced a diving save from Peter Gulacsi.
And Solskjaer's side took the lead Greenwood in the 21st minute. Pogba went on a mazy run forward and timed his pass through to Greenwood to perfection. Greenwood latched on it and dispatched it calmly into the bottom right corner on his first ever start in the competition.
United goalkeeper David de Gea was called in to action for the first time in the match two minutes later. Emil Forsberg moved the ball quickly to Christopher Nkunku, who lashed a high effort towards the near post and De Gea raised his arms high to push it out for a Leipzig corner.
Julian Nagelsmann's side came close in the 65th minute when De Gea leapt to his left to keep out a strong header from Ibrahima Konate after Leipzig won a free-kick in United's half.
The Red Devils doubled their advantage in the 74th minute. Bruno Fernandes found Marcus Rashford with a sumptuous pass. Rashford timed0 his run perfectly, dashing from inside his own half to get one-on-one with Gulacsi and fired into the net.
Rashford grabbed a quick-fire second after Dayot Upamecano's error presented the ball to the striker who absolutely lashed the ball into the back of the net.
The Red Devils had a fourth in the 87th minute. Marcel Sabitzer scythed down Anthony Martial in the box with the referee pointing to the spot. The French striker dusted himself and smashed the penalty in to the bottom left corner.
And Rashford put the cherry on the top when he netted his hat-trick goal in stoppage time. Martial drew in a few Leipzig defenders and laid the ball off to Rashford in the box, with the England international taking a touch before spanking the ball past Gulacsi.