Milan recovered some of the spark they lacked in the first leg but Inter's Martinez put the tie to bed when he struck powerfully from close range in the 74th minute at the San Siro.
Inter went into the game holding a 2-0 advantage after Edin Dzeko and Henrikh Mkhitaryan had struck early in a flying start to the first leg and never looked in serious danger of missing out on the final, where they will face Real Madrid or Manchester City on June 10.
Inter, who overcame Portuguese pair Porto and Benfica to reach the semis, will play in the final for the sixth time and first since lifting their third European Cup in 2009-10 under Jose Mourinho.
Inter coach Simone Inzaghi fielded an unchanged team from the first leg, with Martinez and Dzeko up front, having rested players in Serie A over the weekend.
Main forward Rafael Leao returned for Milan after an adductor problem sidelined him in the first leg, with Junior Messias and Malick Thiaw starting in place of Alexis Saelemaekers and Simon Kjaer.
Inter Milan Lautaro Martinez scores their first goal
A much more daring Milan side could have scored early but Inter keeper Andre Onana saved a low Brahim Diaz shot that was heading towards the bottom-right corner.
Leao had a great chance to halve the deficit late in the first half but struck inches wide, before Milan keeper Mike Maignan superbly denied Dzeko who leaped high to meet a free kick.
AC Milan\'s Sandro Tonali and Olivier Giroud look dejected after the match
While Stefano Pioli's Milan kept probing for an opening to spark a comeback, Inter hinted in the first half that they would not be taking their foot of the gas.
Martinez eventually proved Milan's executioner, playing a one-two with substitute Romelu Lukaku before scoring his sixth goal in Inter's last six games.
It is the first time Inter have progressed from a knockout tie against their city rivals in the competition, with Milan coming out on top both in the 2002-03 semi-finals and the 2004-05 quarters.
Inter Milan coach Simone Inzaghi celebrates with kids after the match
It is also the first time in almost 50 years that Milan have lost to Inter four times in a single campaign, the only other time being in 1973-74.
And never before in their history had they gone four games in a row without finding the net against Inter.
"It's clear that we dreamed of the final and wanted to beat our rivals," Pioli said.
"Not succeeding causes us disappointment, but we'll have to focus on the championship. The real disappointment would be not playing in the Champions League next year."
Milan, who have won only two in their last seven Serie A games, risk dropping out of Europe's elite club competition next season as they currently sit four points off the top four.
Inter Milan fans celebrate after the match