Cape Town - The match was a chance to shine for young United forward Marcus Rashford, who showed his pace early on before testing goalkeeper Mikkel Andersen in the 16th minute, receiving a throw in and dodging the defender on the edge of the box before firing a low effort that needed to be pushed round the post for a corner.
The Danes extended their advantage in the 27th minute through Pione Sisto, who also scored in the first leg. The young Uganda-born Dane finished off a good move by the visitors, getting the better of Daley Blind and Michael Carrick before drilling it low and hard into the net.
United almost pulled one back on the half-hour mark through Memphis Depay’s opportunistic first-time effort, but the Red Devils had their goal just a minute later, and the Dutchman would be the man to set it up.
Nikolay Bodurov was the unlucky scorer for United after Depay got away down the left flank and drilled the ball across goal, and the Bulgarian defender could only find his own net as he attempted to hoof it clear.
United then had a number of chances to score again, with Morgan Schneiderlin volleying wide, Memphis having a shot from long range that curled just wide, and Schneiderlin hitting the post with a header from a corner.
The pressure paid off in the 42nd minute when Ander Herrera was fouled in the box by Andrea Romer, and United were given a penalty. Juan Mata stepped up to the spot, and looked to fire it low into the bottom corner, but Andersen was able to make the save, tipping it round the post.
United continued to press for the aggregate equaliser in the second half, with Schneiderlin and Herrera both missing chances before Sergio Romero was called into action at the other end to deny Paul Onuachu.
Rashford had been enjoying a good game for United on his debut , so it was fitting that the youngster was the one to level the scores on aggregate. The Danes couldn't get the ball clear, allowing Mata to pick out the teenager, who smashed his effort into the net in the 63rd minute.
United then won the tie in the 74th minute, and it was Rashford again with the finish. Mata got down the right side of the box, cut back onto his left foot, and lofted a cross in towards the far post, where Rashford applied the finish.
United were awarded another penalty in the 87th minute when Memphis went for a shot that hit defender Kian Hansen on the arm. It was Herrera, and not Mata, to take the penalty, and this time there was no mistake.
Romer was then shown a second yellow card and given his marching orders as the tie slipped away from the Danes, before Memphis produced a dazzling little dribble and fired into the bottom corner to end the match in style.