The result marked Genoa's first victory of the campaign and left Samp without a win after three rounds of the new Serie A season.
"In the most important game, the encounter that means the most to our fans, we let our supporters and ourselves down," Rossi told Gazzetta dello Sport. "I cannot believe how bad we were, it was unacceptable.
"I apologise to our fans and I personally will say sorry to the club.
"I am the one responsible."
Genoa dominated the game and took a deserved 1-0 half-time lead, courtesy of Luca Antonini's goal.
Samp hardly threatened and gave away two further goals after the re-start from Emanuele Calaio and Francesco Lodi.
"Genoa were the better team," Rossi said. "They did a good job and made the most of their chances.
"They deserved the win."
Samp, who have mustered one point so far, travel to Sardinia to face Cagliari on Saturday and Rossi expects a drastic improvement from his players.
"We have a lot of work to do," he said. "We now need to get our hands dirty in order to find the fighting spirit that we lacked against Genoa."