The visitors got their campaign up and running quickly in Genoa after Diaz, who rejoined Milan on loan from Real Madrid for the next two seasons, squeezed the ball home after nine minutes to break the deadlock.
Both sides more chances in the first half, with new Milan goalkeeper Mike Maignan making several fine stops to preserve his side's lead, although the pace slowed as the match wore on.
Close-season Milan signing Olivier Giroud did have two late opportunities to cap his debut with a goal, but Diaz's strike proved to be enough for Stefano Pioli's side.
"I think we all fight together and on the pitch the spirit is the right one," Diaz told DAZN. "The team always does something more. With this spirit we will do great things.
"Wearing Milan's number 10 shirt is not something I'm afraid of."
With champions Internazionale getting off to a flying start with a 4-0 victory over Sampdoria's local adversaries Genoa on Saturday, Milan had to make a statement of their own, as they chase their first Scudetto in a decade.
Pioli has had to start the campaign without last season's top scorer Zlatan Ibrahamovic, who is out with a knee injury, but the Milan coach was able to blood another veteran striker up front in close-season signing Giroud from Chelsea.
After Diaz had broken the deadlock with a shot goalkeeper Emil Audero should really have kept out, the man tasked by Milan with replacing Italy keeper Gianluigi Donnarumma - Maignan -proved unbeatable as Sampdoria piled the pressure on.
Rade Krunic should have made it two late in the first half, but scuffed his shot wide from close range.
Maignan had to be at his best to deny Manolo Gabbiadini early in the second half but that was close as the hosts got to equalising, with Roberto D'Aversa's side looking shorn of ideas as Milan got men behind the ball to see the victory through.