Higuain had gone nine games without a goal going into Saturday's contest at San Siro, and rarely looked like breaking his duck in a largely unconvincing display.
However, he struck the decisive blow from close range in the 64th minute as Milan, who saw Suso sent off late on, ended a run of five games without a win by coming from a goal down, Samu Castillejo having initially cancelled out Andrea Petagna's opener.
There had been reports Gattuso could be sacked if Milan failed to claim a positive result, but the Rossoneri now head into a two-week break on a high ahead of a Coppa Italia clash with Sampdoria and the Suppercoppa Italiana meeting with Juventus.
And Higuain was quick to pay tribute to his coach after they moved within a point of the Champions League places.
"It was an important goal and three fundamental points after all we've wasted before, but now we can have a rest with our families," the striker told DAZN.
"It's natural the fans expect more from me. I came to Milan because I was convinced by the project here and I want to continue like this.
"Gattuso has always supported me and above all been honest. This goal is for him. Now we've got the Supercoppa, it's the first trophy of the season and we want to win."