Milan, five-time winners of the Cup, will meet either Inter Milan or Lazio in the two-legged semi-final.
Abraham, who joined Milan from Roma in August on a season-long loan deal, put the hosts in front in the 16th minute on Wednesday night before punishing his old teammates again three minutes before the break.
Roma substitute Artem Dovbyk pulled one back nine minutes into the second half, before Felix came off the bench and netted in the 71st minute, two days after his arrival from Stamford Bridge.
After a cagey opening to the tie, Milan struck with their first real chance.
Roma keeper Mile Svilar parried a ferocious Tijjani Reijnders shot, the ball fell to Theo Hernandez who sent a cross back across the area and Abraham hung in the air before steering his header into the far corner.
Milan were clinging to their lead, but struck against the run of play after regaining possession in midfield.
Hernandez drove forward before rolling the ball out to Abraham just outside the box, with time to close in on goal before driving his shot past the onrushing keeper.
"Everybody knows how I feel about Roma, it's a club that means a lot to me. It wasn't nice," Abraham told broadcaster Mediaset after he was booed by the away fans.
"I think they thought I celebrated but I didn't celebrate, I was just thanking God. This is football, it happens."
Roma made three changes at the start of the second half, and that decision paid off when Dovbyk put his side back in the game.
Angelino's cross took a wicked deflection off former Manchester City defender Kyle Walker and Dovbyk was at the back post to poke the ball home.
Milan coach Sergio Conceicao responded by sending on two deadline day signings, Felix and Santiago Gimenez, and the new arrivals combined to double the lead.
Gimenez played the perfect pass through to Felix who beat the offside trap and calmly chipped the ball over Svilar.
Reijnders headed the ball into his own net but Roma's joy was short-lived with the goal chalked off for an offside in the build-up.
The visitors kept on pushing but Milan, winners of the Italian Super Cup last month, remain on target for more silverware.
