Ex-Chelsea forward Abraham opened the scoring early in the second half, firing right-footed across goalkeeper Josep Martinez into the bottom corner.
Hakan Calhanoglu equalised for Inter midway through the second period, against his former club, with a powerful low drive past Mike Maignan from 25 yards.
The second leg is scheduled for 23 April, with the winner facing either Bologna or Empoli in the final on 14 May.
Bologna won 3-0 at Empoli in the first leg of their semi-final on Tuesday.
This was the fourth Milan derby of the 2024-25 season and Serie A leaders Inter have failed to win any despite having 20 points more than their city rivals in the league.
Maignan was the busier of the two goalkeepers in a goalless first half, keeping out efforts from Inter trio Joaquin Correa, Calhanoglu and Davide Frattesi.
Abraham was called upon defensively too – nodding Stefan de Vrij’s goalbound header to safety.
Maignan denied substitute Nicola Zalewski from close range as Inter pressed for a winner in the aftermath of Calhanoglu’s leveller, while at the other end Rafael Leao curled a late effort just wide from the edge of the box for Milan.
