The result leaves Fiorentina without a win in their opening three Serie A fixtures, having already been thrashed by an aggregate score of 7-0 in their first two league outings this season, while Torino picked up a first victory of their own campaign under new head coach Ignazio Abate.
Grosso accepted responsibility for another damaging result in Florence.
"The responsibility is mine. We certainly haven't laid strong foundations to build the rest on. Difficult challenges need time," he said.
"It's important to understand that in this kind of match we had the quality to hold on and see it through to the end, given how we had set it up.
"Some lapses in concentration didn't allow us to do that. This is another disappointment for the city, but also for us — from here we have to start again."
Asked whether the club's dismal start was a source of concern, Grosso did not attempt to hide his unease.
"I'm always worried. Thinking this way when you're part of such a difficult championship is normal, especially with a team that's this difficult and needs time," he said. "This disappointment adds to the first two, and obviously there's disappointment inside."
Abate, meanwhile, was full of praise for Che Adams, whose future at the club had been the subject of transfer speculation over the summer before he stayed and scored his second league goal of the season to settle the contest.
"I spoke to him straight away, telling him I'd be happy if he stayed. He's got strength in his legs, he holds up play and he scores goals. He's important for us," the Torino head coach said. "Then there are some annoying market dynamics, but on our part there has always been a willingness to keep him."
Abate also explained his decision to leave Simeone out of his starting line-up.
"It was a tactical choice — he'd played three matches and I'd seen Adams playing well in training. I wanted to put an extra midfielder on and push Vlasic further forward," he said.
The Torino boss reflected on his side's tactical approach after the interval.
"We had great possession in the first 35-40 minutes, then Fiorentina raised their line, and that's why I decided to switch to a diamond in midfield," Abate said.
"At right-back, Comert had a great game, I'm pleased with Bragança's introduction, and Fitz-Jim showed his work ethic again. But I'm happy with everyone."
The win moved Torino off the foot of the table after a difficult opening to the season under their new head coach.