The visitors were out of the blocks quickly in Genoa as Diaz, who has rejoined Milan on loan from Real Madrid, squeezed the ball home in the ninth minute.
New Milan goalkeeper Mike Maignan, between the posts after Gianluigi Donnarumma’s move to Paris St Germain this summer, made several fine stops to preserve his side’s lead.
Olivier Giroud had a couple of late opportunities to cap his debut with a goal, but Diaz’s strike proved to be enough for Stefano Pioli’s side.
Elsewhere in Italy, Joao Pedro’s second-half double – his second goal from the penalty spot – saw Cagliari battle back from two goals down to draw 2-2 with Spezia, who looked in control after strikes from Emmanuel Gyasi and Simone Bastoni.
Erik Lamela’s injury-time strike gave Sevilla victory at Getafe and moved them to the top of LaLiga.
The summer signing from Tottenham netted in the third minute of added time to give Julen Lopetegui’s team two wins from their first two matches.
Meanwhile, Ruben Garcia saw his penalty saved by Matias Dituro as Osasuna had to settle for a goalless draw at home against Celta Vigo.