Mohamed Salah's early penalty in the return match on Thursday gave Liverpool a 1-0 victory in Italy but Klopp's side went out 3-1 on aggregate.
Pogba, who rejoined United from Juventus last year, started all 15 games and scored three times in Europe's second-tier competition to help United lift the trophy for the first time in May.
Roundup of the matches being held across major football leagues in Europe on Sunday
Arsenal can sustain a Europa League challenge but a top-four finish in the Premier League would be the best way to reach next season's Champions League, manager Arsene Wenger has said.
Manchester City won an unprecedented fourth straight English top flight title with a 3-1 win over West Ham United.
Second-half strikes by Federico Valverde, Brahim Diaz and Endrick gave feeble Real Madrid an unconvincing 3-0 win over lowly Real Valladolid.
Allegri was sent off towards the end of the game for his fierce protests over refereeing decisions.
Bayer Leverkusen won their first trophy in 31 years with five matches left in the season, snapping Bayern Munich's 11-year stranglehold on the Bundesliga.
Liverpool hammered Sparta Prague 6-1 on Thursday to storm into the Europa League quarter-finals 11-2 on aggregate.
UEFA has postponed all Champions League and Europa League matches due to take place next week because of the coronavirus outbreak, European soccer's governing body said in a statement on Friday.
Barcelona made a disappointing start to their Champions League campaign by losing 2-1 to AS Monaco on Thursday.
Liverpool lost the plot in Atalanta defeat, says Klopp
IMAGES from the Europa League matches played on Thursday
IMAGES from the Europa League matches played on Thursday
The win left Chelsea in 11th position, a point and one place behind Newcastle but with a game in hand.
Harry Kane netted his 200th goal for Tottenham Hotspur as they won 3-1 at Ludogorets and AC Milan's 24-match unbeaten run ended
Europe's top club coaches gave their support on Thursday to the much-maligned Europa League.
Two goals from Luis Diaz and one from Mohamed Salah helped Liverpool stroll to a 3-0 victory at Manchester United.
Sevilla clinched a sixth Europa League title with a roller-coaster 3-2 win over Inter Milan after an overhead kick by defender Diego Carlos deflected by the Italian side's top scorer Romelu Lukaku settled a thrilling contest on Friday.
IMAGES from the Europa League semifinal played between Manchester United and Sevilla in Cologne on Sunday.
Liverpool overcame an early scare and came from a goal down to claim a 3-1 win at Austria's LASK Linz.
Pinpoint headed goals from Trevoh Chalobah and Nicolas Jackson earned Chelsea a 2-0 victory over London rivals Tottenham Hotspur in the Premier League on Thursday, dealing a blow to Spurs' hopes of qualifying for next season's Champions League.
Alexandre Lacazette's free-kick sent Arsenal into the Europa League semi-finals with a comfortable 1-0 win away over Napoli, while Chelsea beat Slavia Prague 4-3 in a roller-coaster game, helped by a Pedro brace.
Manchester United manager Jose Mourinho said Friday's draw for the last 16 of the Europa League against Russian side Rostov has been unkind to his team, the favourites.
Arsenal's Pierre-Emerick Aubameyang struck a vital late goal to seal a 3-1 win over Valencia on Thursday and give his side breathing space for the second leg of their Europa League semi-final in Spain next week.
Chelsea will face Dynamo Kiev at home in the first leg of their Europa League last-16 tie, while London rivals Arsenal will travel to Rennes following Friday's draw in Nyon, Switzerland.
AS Roma head coach Jose Mourinho has been suspended for their next four matches in the Europa League.
On Sunday, after Liverpool's final game of the season, a 2-0 home win over Wolverhampton Wanderers, Klopp led the Anfield crowd in a song for the incoming manager.
United captain Jesse Lingard fired them ahead in the 10th minute with a fizzing low shot from 20 metres before Dmitri Shomko and a Di'Shon Bernard own goal turned the match on its head shortly after half-time.
Manchester United, Sevilla and Celtic led a parade of clubs with continental pedigree to book the first places in this season's Europa League knockout stage.
We want to fight for that title and we are very excited to see the draw, says Ander Herrera
Bayer Leverkusen crushed hosts VfL Bochum 5-0 to stretch their remarkable unbeaten run across all competitions this season to 50 consecutive matches.
Arsenal are not only hoping to win a first European trophy since 1994, but know their hopes of qualifying for next year's Champions League depend on winning the Europa League in the French city of Lyon next month.
While Spurs play Inter Milan at home in Champions League on Wednesday, on Thursday, Arsenal face a long trip to Ukraine to play Vorskla Poltava in the Europa League, needing a victory to consolidate first place in the group.
Defender Kolo Toure believes Liverpool can recover from Sunday's painful League Cup final shootout defeat against Manchester City at Wembley and rescue their season by winning the Europa League. Juergen Klopp's side, seeking their first trophy in four years, face bitter rivals Manchester United this month in the last-16 of the continent's second tier competition. "You have to learn to lose sometimes before you win. We learn as a team together. The Europa League can save the season and the Premier League is not over yet," Toure told British media. Liverpool, who are ninth in the table, can quickly avenge Sunday's defeat when they host fourth-placed City in the Premier League on Wednesday. Goalkeeper Simon Mignolet said it was important for the five-times European champions to bounce back quickly from their Wembley disappointment.
Arsenal defender Per Mertesacker says the players are focused on winning the Europa League as well as their four remaining Premier League games to give manager Arsene Wenger the perfect send-off.
Shakhtar dominated possession as expected but were toothless up front and Inter opened the scoring with their first chance after 19 minutes.
Olympique de Marseille were the fall guys on a hectic opening night of Europa League group stage action as German club Eintracht Frankfurt returned from the south of France with a surprise 2-1 victory on Thursday.
After scoring twice in Chelsea's 4-1 win over Arsenal in the Europa League final, Eden Hazard said it was the moment to call time on his magnificent career with the London club to seek new challenges.
IMAGES from the FA Cup final played between Manchester City and Manchester United, at Wembley Stadium, London, Britain, on Saturday