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 looming, there was one final twist.
Carlos flew into the air to meet a high ball with an overhead kick which was probably 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.
Carlos goes from villain to hero as Sevilla continue Europa dominance
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.
Former Spain and Real Madrid boss Lopetegui had never won a trophy in his managerial career and there was a complete overhaul of the squad last summer.
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 he tried an audacious effort with help from Lukaku.