The 18-year-old was only in the side after Anthony Martial had suffered an injury in the warm-up, but he added to a Nikolay Bodurov own goal before late strikes from Ander Herrera and Memphis Depay sealed a 6-3 aggregate victory.
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United went into the match looking to overturn the shock 2-1 deficit from the first leg last week, and it was the late inclusion Rashford who had the first opening when he cut inside from the left channel before drilling a low strike towards goal that was turned behind by Mikkel Andersen.
For the most part, though, Andersen was well protected by the Midtjylland defence during the opening stages despite United dominating possession, with the hosts' only other threatening moment coming when Filip Novak's clearance bounced off Jesse Lingard into the side-netting.
It was the visitors who were to take a shock lead shortly before the half-hour mark when Pione Sisto made their first meaningful attack of the game count by showing composure in the area to beat two defenders and pick out the bottom corner.
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The goal left United needing three without reply to progress without the need for penalties, and they immediately set about looking for them as Depay sent a curling effort into the arms of the keeper.
It didn't take them long to level things up on the night as Depay once again made good progress down the left flank, this time going on the outside and firing a low cross into the box that Bodurov could only turn into his own net.
That only served to encourage the hosts more, and Morgan Schneiderlin came close with an acrobatic bicycle kick before Depay curled a long-range strike wide of the near post as United continued to ramp up the pressure.
Schneiderlin came even closer to getting his name on the scoresheet when he planted a header against the post, but it looked as though that miss would not come back to haunt his side when they were awarded a penalty seconds later.
Romer was the man penalised for a rash challenge on Herrera just inside the area, but Andersen came to his side's rescue when turning Juan Mata's resulting spot kick around the post.
Despite that setback, United started the second half quickly and missed two clear chances in quick succession when first Schneiderlin and then Herrera failed to hit the target having been found unmarked at the back post.
United's second goal finally arrived shortly after the hour mark, and it was Rashford who marked his debut with the crucial strike as he reacted quickest to sweep Mata's cutback home.
Things got even better for the youngster as he doubled his personal tally with 15 minutes remaining, again drifting away from his marker to plant a well-controlled finish past the keeper after being found at the back post by Guillermo Varela.
One goal for Midtjylland would have restored their aggregate lead, but United put the tie to bed in the 87th minute when they were awarded a second spot kick, which Herrera duly dispatched having replaced Mata on penalty duties.
Midtjylland's Romer then capped off a forgettable individual display by receiving a late red card, and Depay took full advantage to drive home a fifth for United moments later, ending a man-of-the-match display with a goal and sealing a 6-3 aggregate victory.