The advantage was short-lived, though, as Luuk de Jong's bullet header in the 12th minute brought Sevilla level only for the Netherlands international to head home from an Ever Banega free kick a little over 20 minutes later to leave Inter trailing 2-1.
A rip-roaring first half ended with Lucas Ocampos shaving the bar with a header as both sides surged forward in relentless end-to-end action and headed to the dressing room at half-time locked in a 2-2 draw.
The scoring pace slowed in the second half and Lukaku saw a golden chance to put Inter back in front from a breakaway in the 65th minute denied by an onrushing Bono to keep the score level.
Carlos then went from villain to hero in the 74th minute when his overhead kick following a set piece took a deflection off of Lukaku and wound up in the back of the net to give Sevilla a 3-2 lead.
"This group deserve this. We've been fighting all season and getting over the problems that we've encountered," said Sevilla right-back Jesus Navas, who won his third Europa League with Sevilla -- 14 years after his first.
The loss extended Inter's wait for a first European trophy since Jose Mourinho led the Italian club to the Champions League title in 2010. Sevilla have won the Europa League in four of the last seven seasons.
"I think Sevilla were more tired than us in the final part of the game but they had this chance, the ball was deflected by Romelu's foot and went in. It happens," goalkeeper Samir Handanovic said.
It was a joyful end of the season for Sevilla coach Julen Lopetegui, who won his first trophy as coach at senior level after a torrid period which saw him sacked by Spain and Real Madrid.
"You have to know how to cope with difficult moments and overcome them. This is a feeling of immense happiness," Lopetegui said.
"I'm so grateful to our players for how hard they worked. Our anthem says we never give up and tonight we proved that again."
In the other dugout, Conte was coy about his future at the Italian side, saying that he will take some days off before meeting with the club's ownership, although he did infer he could exit.
"In a serene way we will try to plan the future of Inter with or without me, we'll see ...There is no resentment, there are perhaps different views and evaluations will be made."
The head coach's occasional angry outbursts have seen him clash with Inter's senior executives since taking over the reins in 2019.
"We'll clarify the situation, without rancour, because I will always be grateful for having the chance to spend a great year at Inter, but on the other it was also very tough."