United had been trailing 2-1 from the first leg, and went 3-1 down on aggregate to the Danish champions before an own-goal, 18-year-old Rashford’s brace, an emphatic Ander Herrera penalty and a late Memphis Depay strike saw United qualify 6-3 on aggregate.
Tottenham started their game on a calmer note having drawn the first game in Florence 1-1, and eventually eased past the Italian’s 4-1 on aggregate.
Ryan Mason curled home a peach of an opener after being brilliantly set up by England midfielder Dele Alli on 25 minutes while Argentine Erik Lamela hammered home in a goalmouth scramble to make it 2-0 just after the hour.
An own-goal from Fiorentina’s Gonzalo Rodriguez completed the Italians’ misery.
An early James Milner penalty saw nervy Liverpool beat Augsburg 1-0 as Jurgen Klopp’s men also went into the hat for the last-16 on Friday.
United’s best player on the night was undoubtedly Memphis Depay, but Rashford, a graduate from United’s youth team, stole the headlines after getting his start after Anthony Martial picked up a hamstring injury in the warm up.
He first clipped home Juan Mata’s cut back from around the penalty spot on 64 minutes and from a similar position ten minutes later then tapped home from Guillermo Varela’s cross, giving United the lead for the first time in the contest.
“We’ve seen a special debut and hopefully his career wil kick off from here,” former United player Paul Scholes told BT Sport.
“There’s some very good teams in this still, so it isn’t going to be easy.”
The last-32 tie at Anfield had been finely poised after last week’s first leg in Bavaria ended goalless and the home side also had one eye on Sunday’s League Cup final at Wembley against Manchester City.
Misfiring Liverpool eventually had to survive some late scares, notably when Augsburg left-back Kostas Stalfydis sent a free kick inches wide with the clock ticking down.
“If you can’t decide a game early enough it’s always nervous,” said Klopp.
“It was a really good game and it everyone who saw it would say we deserved to win.
Milner’s slick penalty conversion on five minutes came after Augsburg’s Dominik Kohr handled after his arm got into a tangle with a teammate leaping to head away a shot.
Tournament favourites Napoli could only draw 1-1 with Spaniards Villarreal and crashed out 2-1 while Borussia Dortmund beat Porto 1-0 to ease through 3-0 over the two legs.
Gary Neville’s Valencia racked up a 10-0 aggregate score with a 4-0 win at Rapid Vienna, Shakhtar Donetsk stunned Schalke 04 with a 3-0 win in Germany while a Karim Bellarabi double for Bayer Leverkusen helped them beat Sporting Lisbon 3-1 on the night and 4-1 on aggregate.
Athletic Bilbao edged Marseille 2-1 after a 1-1 draw in Spain and Lazio beat Galatasaray 3-1 (4-1 on aggregate) to also go into the hat for Friday’s last 16 draw.