The tie was delicately poised at 2-2 following the first leg at Old Trafford, but Sevilla were comfortably the superior side on the night against a dire Man United.
Youssef En-Nesyri opened the scoring in the eighth minute of the match before Loic Bade doubled Sevilla's advantage early in the second half; En-Nesyri then scored a second in the latter stages, with Jose Luis Mendilibar's side winning 3-0 on the night to advance to the semi-finals of the competition with a 5-2 aggregate success.
Los Nervionenses are the most successful side in the history of this competition with six titles, and they will now take on Juventus in the final four next month.
Man United, meanwhile, must now claim a top-four spot in this season's Premier League table if they are to be present in the 2023-24 Champions League, and they are currently third, six points clear of fifth-placed Tottenham Hotspur with a game in hand.
Ten Hag made just one change to the side that started against Nottingham Forest in the Premier League on Sunday, with fit-again Marcel Sabitzer coming in for the suspended Bruno Fernandes.
Sevilla's main team news involved their centre-forward, with leading goalscorer En-Nesyri coming in to lead the attack after surprisingly being named on the bench for the first leg at Old Trafford.
There was a red-hot atmosphere in the early stages of the contest, and Sevilla took the lead in the eighth minute of the contest following a mistake from Harry Maguire, who gave the ball away to Erik Lamela in a dangerous area, and it bounced for En-Nesyri, who expertly found the bottom corner.
Maguire called for the pass from De Gea, and the goalkeeper obliged, but the England international could not sort out his feet, with the Red Devils making the worst possible start at Estadio Ramon.
Ivan Rakitic had another opportunity from a free kick in the 15th minute following a poor challenge from Aaron Wan-Bissaka, but his effort was straight into the wall.
Wan-Bissaka had Man United's first opportunity in the 23rd minute when the ball broke for the right-back inside the Sevilla box, but his effort was straight at Sevilla goalkeeper Bono.
Sevilla threatened again in the 27th minute, with En-Nesyri heading down towards Lucas Ocampos inside the Man United box, but De Gea just about managed to claim under pressure, with Diogo Dalot also playing his part.
Casemiro had a half-chance in the 36th minute when Antony picked out his teammate with a deep cross, but the midfielder headed over the crossbar from a dangerous position.
De Gea had to keep down to keep out a strike from Lamela in the 38th minute, before Suso had an effort blocked by Victor Lindelof during a difficult period for the away team.
Sevilla thought that they had scored their second of the match in the 40th minute, with Ocampos picking out the back of the net with a brilliant finish, but the goal was disallowed, with Marcos Acuna in an offside position.
Man United had a shout for a penalty in the 43rd minute when Jadon Sancho went down inside the box under a challenge from Fernando, but the referee was not interested and VAR agreed, with replays showing that the ball had been played by the experienced midfielder.
Dalot did well to turn a low Ocampos cross wide of the post in the closing stages of the first period before a thunderous volley from Rakitic was deflected past the goal by Casemiro, with Man United incredibly fortunate to only be one goal behind at the half-time break.
Ten Hag made a double change at the break, with Marcus Rashford and Luke Shaw both entering the field after recovering from injuries to make the squad; Sancho and Wan-Bissaka were the players to exit, with both struggling in the opening 45 minutes of action.
Sevilla scored a second right at the start of the second period, though, with Bade converting a corner from Rakitic into the back of the net off his shoulder; the ball hit the underside of the crossbar before dropping over the line, with De Gea caught out in another questionable moment from the Man United goalkeeper.
De Gea pushed a curling Rakitic effort wide of the post in the 51st minute, before Man United somehow survived from the resulting corner, with De Gea keeping out a close-range Ocampos effort.
Wout Weghorst was introduced for the Red Devils in the spell that followed, with Martial limping off, and the Dutchman almost released Rashford with a pass in the 56th minute, but it was just too strong.
Bono had to keep out a strike from Casemiro in the 62nd minute, before Weghorst had a shot blocked, with Man United starting to find some space in the final third of the field.
Christian Eriksen had a free kick deflected just wide of the goal in the 72nd minute, before Casemiro missed the far post with an effort seven minutes later, with Sevilla still in control of the tie.
De Gea had another horror moment in the 81st minute, with the Man United goalkeeper leaving his goal and miskicking the ball straight to En-Nesyri, who fired into an empty net to end any hopes of a late comeback for the 20-time English champions.
Anthony Elanga tested Bono with a late strike, but it was a horror performance from the visitors, who never came close to making it an uncomfortable night for Sevilla.
Man United will now switch their attention to the FA Cup, preparing to take on Brighton & Hove Albion in the semi-finals of the competition on Sunday afternoon.
Sevilla, meanwhile, will resume their La Liga campaign at home to Villarreal on Sunday night, with Mendilibar's side bidding to build on their 2-0 success over Valencia in the competition last time out.