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.
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.
"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."