Veteran centre-forward Ibrahimovic punished the poor positioning of a Genoa wall, firing in an early free-kick to settle any Milan nerves.
Messias then scored twice, heading past Salvatore Sirigu in first-half stoppage time before fizzing a low strike into the bottom corner just after the hour mark.
Genoa were hardly able to threaten the visitors, with Milan's dominance duly reflected in the scoreline as the Rossoneri moved to within a point of leaders Napoli.
Ibrahimovic opened the scoring in just the 10th minute, bending his free-kick around the centrally positioned wall and into the bottom-right corner, leaving Sirigu with no chance in the Genoa goal.
The 40-year-old started the game in lively fashion, taking six of Milan's first seven shots in the opening 20 minutes, but it was Brahim Diaz who came closest to doubling the visitors' advantage, lashing narrowly over after a superb solo run shortly before the break.
Milan did go 2-0 up soon after, with Messias powering a looping header across goal and over the diving Sirigu after Rade Krunic's shot in the box was blocked and popped up into the air.
Hernani came close to pulling one back for Genoa early in the second half, but his towering header was kept out spectacularly by Mike Maignan before Fikayo Tomori produced a spectacular bicycle-kick clearance off the line.
It was Milan that scored next, Messias steering a low effort into the bottom-left corner to round off a flowing team move and give his side a three-goal cushion, which they held comfortably through to the final whistle.