Milan came into the game looking for an improvement after back-to-back league losses to Fiorentina and Sassuolo, and they took control early on at the Stadio Luigi Ferraris when Zlatan Ibrahimovic fired home a 10th-minute free kick.
"Today I did the best I could. These guys make me enjoy myself and make me feel young," Ibrahimovic told Sky Italia.
"Unfortunately we lost the last two games but the season is still long. There are ups and downs over the year."
Brazilian winger Messias, who scored the winner against Atletico Madrid on his Champions League debut last week, was named in Stefano Pioli's starting line-up for the first time and marked the occasion in style by scoring either side of halftime.
Messias continues a remarkable career journey that has gone from delivering fridges to now becoming a star in Milan. The 30-year-old joined the Rossoneri from Crotone in the off-season. Messias moved to Italy from Brazil in 2011 and initially juggled amateur football with a job delivering fridges, before slowly climbing up the football ladder.
Shevchenko, who won the Champions League and Serie A with the Rossoneri as a player, is still waiting for his first win as Genoa coach after three matches in charge, and his side are 18th with 10 points.
Milan were under pressure to get a result after picking up one point from their last three Serie A games, and seeing Inter Milan overtake them earlier in the evening by beating Spezia.
Things started badly for the Rossoneri when defender Simon Kjaer was forced off with injury, but soon improved when Ibrahimovic bent a free kick around the wall for his sixth league goal of the season.
Messias pounced on a deflected shot to send a looping header beyond Salvatore Sirigu, but Genoa were denied a goal by a spectacular Mike Maignan save after the break, as the Frenchman sprang to his left to get a hand to a Caleb Ekuban header.
Brahim Diaz teed up Messias in the box for his second of the evening, a neatly placed finish into the bottom corner, but Maignan had to be alert again late on to tip over an audacious long-range Manolo Portanovo effort.