The drama started when Sevilla defender Diego Carlos fouled Romelu Lukaku for an Inter penalty - and ended when Lukaku deflected Carlos' bicycle kick into his own net.
It means Inter Milan's wait for a trophy will tick over into a 10th year at least.
Lukaku gave the Italian side an early lead from the penalty spot for his 34th goal of the season, extending his record of scoring in consecutive Europa League games to 11.
Luuk de Jong scored two headers - from Jesus Navas and Ever Banega crosses - to put Sevilla into the lead.
Inter were level when Diego Godin nodded in to make it 2-2 in the 35th minute.
The game could not continue at that high tempo, but with extra-time on the cards, there was a twist to come.
Carlos flew into the air to meet a high ball with an overhead kick which was going wide until Lukaku stuck out a leg and diverted it past his own goalkeeper.
It was a first trophy in the managerial career of Sevilla boss Julen Lopetegui, whose side are unbeaten since February, a run of 21 games.
"This is a magnificent club," said the former Spain and Real Madrid boss. "I hope all the fans can celebrate it but with caution.
"This squad are great. They believe in it, what they work on and what they transmit. Even if we hadn't won I'd still love these guys, but today we take home the prize. We always believed."
The last nine Europa Leagues all have been won by Spanish or English teams - Sevilla have won four, Chelsea two and Atletico Madrid and Manchester United one each. The last Italian team to win were Parma in 1999.
De Jong is the hero as Sevilla continue Europa dominance
Diego Carlos (r) has scored two goals for Sevilla since joining last summer. The winner was initially awarded to him but later became a Romelu Lukaku own goal
Sevilla's excellent record of never losing a Uefa Cup or Europa League final continues - six finals, six victories.
But this was a new Sevilla team - with barely anybody left since their three wins in a row from 2014 to 2016.
Lopetegui took over last summer and built a new-look team. Nine of the starting XI in Cologne were in their first season at the club.
They were cheered on by two fans, with Sevilla giving two of their 25 tickets for officials to supporters with membership numbers 2108 and 2020 (to make the date of this game).
Carlos looked as if he was going to the villain of the run as he lost Lukaku and then fouled the Belgian striker on the corner of the box after five minutes.
That was the third Europa League game in a row that he conceded a spot-kick, doing the same in wins over Wolves and Manchester United in the previous two rounds.
But their unlikely hero was De Jong, a man who had scored only eight goals since joining last summer - and had a goalless spell with Newcastle in 2014.
His first header was a diving one from a Navas cross and his second was a wonderful looping effort into the top corner from Banega's free-kick.
Carlos' redemption was complete when his audacious effort led to Lukaku's own goal.
Conte's ex-Premier League men fall just short
Romelu Lukaku's deflection helped Sevilla score the winner
Former Chelsea boss Antonio Conte - who has brought in five Premier League players since taking over last summer - led Inter Milan to second place in Serie A this season.
That was their best finish since 2010-11, which was also the last time they won a trophy - the Coppa Italia.
Left wing-back Ashley Young became the first Englishman to start a European final for a non-British club since Owen Hargreaves in the 2001 Champions League final for Bayern Munich.
And it was his former United team-mate Lukaku who had a quick impact when he was fouled for a penalty which he fired home excellently.
The plaudits rolled in at that moment. He is the first player to score 34 goals in his first season with Inter since Ronaldo in 1997-98 and his nine European goals is a joint record for an Inter player in a season.
But he was not the story of the game - in the way he would have wanted anyway - as Sevilla came back.
Inter's confidence rose again when Godin, another big-name signing last summer, headed in.
But Lukaku missed a one-on-one and substitute Alexis Sanchez had a shot cleared off the line, either side of the own-goal winner.
Conte's future is in doubt after disagreements with the club's owners. A trophy would have helped.
"We'll take a few days off and then meet [with the president] with cool heads," he said.
"I don't have many regrets now, my players gave really everything they had against a team used to playing these kind of games.
"Many young players earned a great dose of valuable experience this season. They can continue on this path with great confidence."
Uefa Cup/Europa League finalsSevillaInter Milan2006 - 4-0 win over Middlesbrough1991 - beat Roma 2-1 on aggregate2007 - penalty shootout win over Espanyol after 2-2 draw1994 - beat Austria Salzburg 2-0 on aggregate2014 - penalty shootout win over Benfica after 0-0 draw1997 - lost to Schalke on penalties after 1-1 aggregate draw2015 - 3-2 win over Dnipro1998 - beat Lazio 3-02016 - 3-1 win over Liverpool2020 - 3-2 defeat by Sevilla2020 - 3-2 win over Inter Milan
Match stats - Godin joins select group
Inter lost in a major European final for the first time since the 1997 Uefa Cup, when they lost on penalties to Schalke.
Since 2009-10, the Europa League has been won by a Spanish manager in six of the 11 finals (Unai Emery x3, Quique Sanchez Flores, Rafael Benitez, Julen Lopetegui).
Sevilla have either progressed to the next round or won the competition in 21 of their last 22 Europa League knockout ties.
Inter defender Diego Godin became the sixth player to score in a European Cup/Champions League and Uefa Cup/Europa League final, after Steven Gerrard, Hernan Crespo, Dmitri Alenichev, Allan Simonsen and Pedro.
Romelu Lukaku's goal in the fifth minute for Inter was the earliest in a Uefa Cup/Europa League final since 2001, when Liverpool's Markus Babbel scored in the third minute against Alaves.
Sevilla's Diego Carlos conceded three penalties in the 2019-20 Europa League, the first player to concede that many in a single campaign since Casar Azpilicueta in 2012-13 for Chelsea.
Home TeamSevillaAway TeamInter Milan
PossessionHome48%Away52%
ShotsHome14Away9
Shots on TargetHome5Away5
CornersHome2Away2
FoulsHome16Away19
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SATURDAY 22ND AUGUST 2020
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