Paulo Dybala was on the scoresheet when these two sides met in December's Serie A encounter, and the Argentinian came close to repeating the feat inside 10 minutes, but his powerful header following a corner routine went straight at Mike Maignan.
Minutes later, attention turned to the Roma striker on loan at the home side, when Abraham gave Milan the lead. A Tijjani Reijnders shot was parried away to Theo Hernandez, whose cross was met by the Englishman who planted a header into the bottom corner.
The heading battle continued through much of the first half - Eldor Shomurodov and Fikayo Tomori failed to keep close-range efforts down at either end before Niccolo Pisilli diverted an Angelino cross onto the bar after half an hour.
That signalled a sustained spell of pressure for Claudio Ranieri’s men, but their efforts to find an equaliser before half-time were scuppered as Milan caught them on the break.
Hernandez was again the provider by playing in Abraham, who steered his eighth strike in all competitions past Mile Svilar.
Svilar fared better within minutes of the restart, tipping a Kyle Walker header over the bar and denying the full-back his first goal in Italy.
Walker’s luck remained out soon after when his deflection on Angelino’s centre diverted the ball into the path of Artem Dovbyk, who tapped home to reduce the deficit within 10 minutes of coming on.
In response, Milan twice had the sections of their fans on their feet, only to realise Hernandez’s free-kick had flashed just wide and a tidy Alex Jimenez finish was disallowed for offside.
But where Jimenez failed, debutants Santiago Gimenez and Felix succeeded inside the final 20 minutes - Felix beat the offside trap as Gimenez slotted the ball through, and the Portuguese lifted the ball over the advancing Svilar.
I Giallorossi thought they had closed the gap back to one within minutes when Reijnders turned the ball into his own net, but VAR spared his blushes, spotting an offside against Dovbyk in the build-up.
An 11th successive home match unbeaten for Sergio Conceicao’s side retains hope of just a second Coppa Italia title since the 1970s and ends Roma’s eight-match unbeaten run in domestic competitions.