The Red Devils beat 10-man Real Sociedad 4-1 at Old Trafford for a 5-2 aggregate win, with captain Bruno Fernandes converting two penalties and completing a hat-trick from open play.
United's quarter-final opponents are Olympique Lyon, whose Georges Mikautadze and Ernest Nuamah got two goals each in a 4-0 demolition of Romania's FCSB for a 7-1 aggregate success.
Ange Postecoglou's men defeated Alkmaar 3-1 at London Stadium for a tough 3-2 win in the tie, with two goals from Wilson Odobert and another from James Maddison.
Spurs next face 2022 champions Eintracht Frankfurt, who completed a 6-1 aggregate success over Ajax with a 4-1 home win which included a brace from German 2014 World Cup hero Mario Götze.
With both English clubs languishing closer to the bottom of the table rather than the top in the Premier League, this is the last chance to win a trophy for under pressure managers Postecoglou and Ruben Amorim.
They are joined in the quarter-finals by Athletic Bilbao, Lazio and Bodø/Glimt, who progressed earlier on Thursday.
Rangers became the final team to advance after beating Jose Mourinho's Fenerbahce 3-2 on penalties after the Turkish outfit won the second leg 2-0 in Glasgow to level the tie at 3-3. Rangers keeper Jack Butland was the hero of the shoot-out, making two saves.
Bilbao remained on course towards a Europa League final at their home stadium with a 3-1 victory over 10-man Roma to reach the quarter-finals.
Bilbao overturned a 2-1 deficit from the first leg in Rome from Nico Williams' brace and Yuri Berchiche.
Roma had former Germany defender Mats Hummels sent off in the 11th minute for a last-man foul on Maroan Sannadi, and Leandro Paredes' stoppage time penalty came too late for them.
Frankfurt followed up on their 2-1 success at Ajax last week with a 4-1 home victory over the Amsterdam side which included a brace from German 2014 World Cup hero Mario Götze, and goals from Jean-Matteo Bahoya and Hugo Ekitike.
Bodø/Glimt reached a European quarter-final for a second time, 4-2 on aggregate over reigning Conference League champions Olympiacos.
Coming off a 3-0 home triumph, the Norwegians could afford a 2-1 defeat from Roman Yaremchuk's brace in Piraeus where the hosts had goalkeeper Konstantinos Tzolakis sent off and Rodinei failing to convert a penalty.
Lazio won their last 16 tie 3-2 over Viktoria Plzen from a 1-1 home draw.
In the quarter-finals, Bilbao play either Rangers or Jose Mourinho's Fenerbahce, Frankfurt face Tottenham Hotspur or another Dutch opponent in AZ Alkmaar, while Bodø/Glimt are up against Lazio.
In the Conference League, Chelsea beat Copenhagen 1-0 at Stamford Bridge to progress 3-1 with a goal from Kiernan Dewsbury-Hall in the 55th minute.
Other notable teams to advance were Fiorentina, who beat 10-man Panathinakos 3-1 (5-4 on aggregate), and Real Betis who thumped Vitoria 4-0 away for a 6-2 win in the tie.
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