Donnarumma saved a 50th-minute penalty from Camillo Ciano as part of an electric start to the second half at San Siro.
Krzysztof Piatek and Suso went on to score to wrap up the points for the Rossoneri, whose battle for fourth place in Serie A will go down to the final round of games.
"It was a difficult game, we knew it would be and that Frosinone would come here with the right attitude," Gattuso told Sky Sport Italia
"We had a scare, we thank Gianluigi Donnarumma for getting us out of trouble and can look to the next game."
"In my view, there has been a lot of chatter and I've been considered on the way out since pre-season," he added.
"The table talks, look over the last six years at how many points Milan earned and look where we are now.
"I think we did what we had to do. Perhaps we fell away during the period where we had to finish the job and it dragged on longer than expected.
"Those who need to judge me will judge me. I'll continue this work, I hope to do it at Milan, but either way I think I'm pretty decent and can keep learning. I can do better."