Nicolo Barella put Inter ahead after 15 minutes, driving the ball home after Lukaku's shot was blocked, and six minutes later the Belgian got on the scoresheet when he displayed his strength by holding off defender Edmond Tapsoba before firing home.
Leverkusen winger Kai Havertz, reported to be a transfer target of Premier League club Chelsea, pulled a goal back for the Germans but it was goalkeeper Lukas Hradecky who kept them in the game with a string of fine saves.
"I'm very satisfied because we played exactly how we planned the game, never allowing Leverkusen to play calmly with the ball, always with a certain level of pressure. A coach can only be happy when he sees such commitment," said Inter Milan coach Antonio Conte.
After scoring what proved to be the winner, Lukaku could have had a second goal when Inter were awarded a penalty for handball, but that harsh decision against Leverkusen was overturned by the Video Assistant Referee (VAR).
Tapsoba struggled all night to contain the physicality of 27-year-old Lukaku, who set a record by scoring in nine straight Europa League games, and in the second half the centre-back resorted to hacking the striker down on a number of occasions.
Seeking their first trophy since a 2011 Coppa Italia triumph, Inter sent on Alexis Sanchez in the second half and he had two excellent efforts blocked by last-ditch defending.
Lukaku was again denied a spot kick by VAR after an Inter player was discovered to have handled just before Christian Eriksen was bundled over in the box, and he fired a late shot wide under pressure from Tapsoba as Inter held on for victory.
Lukaku showed plenty of humility after his dominant performance, saying: "To me, the man of the match is Nicolo Barella. The way he plays football is simply incredible -- maturity, quality. He is truly a great player."
Lukaku added: "The defence did a tremendous job again tonight. We are growing but we have to learn to kill games off when we have the chances to do it, and we didn't tonight.
"Now we have time to recover physically and prepare for another great game. We will have to be ready."