The veteran striker scored 11 times in 20 appearances after signing a short-term deal with the Rossoneri last term and he looked sharp in his first competitive outing for this campaign in Dublin.
Ibrahimovic fired the Serie A side into a 23rd-minute lead when he collected Hakan Calhanoglu's clever pass.
Calhanoglu perhaps outshone even Ibrahimovic on the night, though, and he got the goal his performance deserved midway through the second period to make sure of Milan's progress to a home tie with Norwegian side Bodo/Glimt.
Ibrahimovic ended the previous season with five goals in his last four appearances and he looked in the mood early when heading Calhanoglu's free-kick over.
Jack Byrne, Shamrock's own dangerman, then threaded a brilliant pass through to Aaron Greene, whose low effort was kept out by Gianluigi Donnarumma, but Ibrahimovic would soon open his account.
Defender Roberto Lopes failed to clear his lines and Theo Hernandez and Calhanoglu were both involved in Milan's build-up, the latter turning the ball around the corner for Ibrahimovic to produce a clinical finish.
Calhanoglu looked determined to double his side's lead in the second period, producing a bicycle kick that came back off the crossbar and then forcing Alan Mannus into a sharp save with a vicious free-kick.
Eventually he did find a way through when brilliantly picking out the top-right corner of the net following Alexis Saelemaekers' lay-off to make sure of the win.