Inter had a 2-0 lead from the first leg of the semi-final and Lautaro Martinez's 74th-minute goal ended any hopes for Milan to stage a comeback.
Several players from both teams fell to the ground in tears at the final whistle as the Nerazzurri won the so-called "Euroderby" 3-0 on aggregate to progress to their first final since they won the league, Italian Cup and Champions League under José Mourinho in 2010.
Inter players and staff celebrated for long afterwards with their fans, singing and dancing in front of the Curva Nord packed full of Nerazzurri supporters in an electric atmosphere at San Siro - the stadium that both teams share.
"It's obviously a great emotion, a dream that I had together with the guys, that we nurtured right from when the draw was made," Inter coach Stefano Inzaghi said.
"We had to believe in ourselves and we have got here by merit, no one has gifted us anything."
Inter will face either 14-time champions Real Madrid or another juggernaut in Manchester City on June 10 in Istanbul. The second leg of the other semi-final is on Wednesday, with the scoreline locked at 1-1.
"Whoever we end up with, we will be unlucky, because they are two really great teams with incredible quality," Inzaghi said with a wry smile. "I will watch tomorrow's match, like I watched the first leg. … Obviously we will follow it closely."
For Milan, the lengthy wait goes on. They last reached the showpiece event in 2007, when they won the last of their seven titles.
Milan were able to welcome back Rafael Leão. He missed last week's loss with a thigh injury and the Rossoneri were hoping his comeback would inspire the team to do the same.
The difference was immediately apparent as Milan played with an intensity they sorely lacked in the first leg amid a frantic start from both teams.
Brahim Díaz saw a weak shot comfortably smothered by Inter goalkeeper André Onana early on before Leão almost got Milan back into it with his first real sight of goal, shortly before halftime. The winger sprinted into the left of the area but his angled drive grazed the outside of the far post.
Inter also had their chances to score the goal that would have all but killed the tie off as Henrikh Mkhitaryan fired over the bar, while Milan goalkeeper Mike Maignan pulled off a brilliant reaction save to keep out an Edin Džeko header from close range.
Inter sealed their place in the final when Martínez made his way into left side of the area and exchanged passes with Romelu Lukaku - who had come on for Džeko less than 10 minutes earlier - before firing in at the near post.
It was that same formidable attacking partnership that fired Inter to the Serie A title two years ago, earning the duo the nickname "LuLa."
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