After claiming a 2-0 first-leg triumph last week, Inter secured a slender 1-0 win in the second leg, with Lautaro Martinez netting the decisive goal on the 74-minute mark.
The Nerazzurri managed to comfortably see out the contest to book their place in their first Champions League final since winning the competition in 2009-10, and they will face either Manchester City or holders Real Madrid in the showpiece event in Istanbul on June 10.
Simone Inzaghi's men found themselves two goals ahead inside the first 11 minutes of the first leg, and they made another aggressive start in the reverse fixture in front of a cauldron of blue and black, with a fierce fourth-minute volley from Edin Dzeko, who scored in the first leg, palmed away by Milan goalkeeper Mike Maignan; the linesman's flag was raised for an offside in the build-up, though.
Stefano Pioli's side responded well and an audacious 35-yard strike from Theo Hernandez almost caught out Andre Onana, who was rooted to the turf as he watched the shot swerve marginally over the crossbar.
Both goalkeepers were then called into action to make important saves, with Onana firstly keeping out a side-footed effort from Brahim Diaz, before a close-range shot from Nicolo Barella was kept out by a strong left hand of Maignan.
Inzaghi's side were content to sit tight, pounce on the loose balls and hit their rivals on the counter-attack, while the Rossoneri had the greater share of possession throughout the first half, but struggled to break down Inter's robust backline and get star man Rafael Leao into the game.
However, a player of Leao's quality needs few chances to have an impact on a game, and he came agonisingly close to scoring Milan's first goal of the tie when he steamed past Matteo Darmian into the penalty area before fizzing a low-driven effort narrowly wide of the far post.
Inter then came close at the other end when a dangerous free kick from Hakan Calhanoglu was flicked on by Martinez and brilliantly saved at point-blank range from Maignan, before Fikayo Tomori booted the ball away from danger.
The Nerazzurri were dealt an injury blow just a few minutes before the break as Henrikh Mkhitaryan, who scored in the first leg, was forced off after rolling his ankle, with fellow experienced midfielder Marcelo Brozovic brought on as his replacement.
Four minutes after half time, Francesco Acerbi was somewhat fortunate to not be cautioned for an apparent stamp on Sandro Tonali inside Milan's penalty area. VAR looked at the incident but decided that no punishment should be awarded to the Inter centre-back.
While Milan seemed to lack ideas in the final third, Inter comfortably controlled proceedings in the second half and eventually broke the deadlock on the night in the 74th minute, courtesy of a brilliant link up between Martinez and substitute Romelu Lukaku.
Shortly after coming on for Dzeko, Lukaku toyed with the Milan defenders inside the penalty box before expertly threading a pass towards Martinez, who managed to squeeze a low-driven shot beyond Maignan at his near post, sparking wild scenes among the Inter faithful.
Martinez, a World Cup winner with Argentina in December, came close to doubling his tally just a couple of minutes later, but his attempted lob of Maignan was kept out by the back-pedalling French shot-stopper.
Milan have never previously endured four successive matches without scoring against their arch rivals in their entire history, but once again they lacked a cutting edge in the final third and failed to put Inter's stubborn backline under any real danger.
Inzaghi's side have four more domestic fixtures to prepare for before turning their attention to next month's Champions League final, including Sunday's Serie A trip to champions Napoli and next Wednesday's Coppa Italia final against Fiorentina.
Milan, meanwhile, will endeavour to keep their top-four hopes alive when they welcome already-relegated Sampdoria to San Siro for their next league game on Saturday.No Data Analysis info