The fiery former midfielder had reportedly been given two matches to save his job after successive defeats by Inter Milan in the league and Real Betis in the Europa League last week.
"I wanted lions, I wanted angry dogs and that's what we were," Gattuso, appointed last November, told reporters.
"These lads always give their all although sometimes the results don't come but today we did very well against a strong side, in terms of character, personality and our mentality."
Milan took a 17th minute lead when Patrick Cutrone headed in Suso's cross unmarked at the far post.
But within 13 minutes the hosts were 2-1 behind as Riccardo Saponara and Fabio Quagliarella twice combined for Samp.
Quagliarella produced a brilliant turn before chipping a pass to Saponara who cut inside and rolled a shot inside the far post. Then Saponara returned the favour by lofting a diagonal ball over the Milan defence into the path of Quagliarella and the 37-year-old met it in his stride to volley past Gianluigi Donnarumma. However, Gonzalo Higuain levelled for Milan before halftime after playing a one-two with Cutrone inside the penalty area and scoring from a narrow angle.
Suso settled the match with a trademark goal in the 62nd minute when he collected a pass on the touchline, cut inside and scored with a shot which bounced in front of keeper Emil Audero.
Milan are fifth with 15 points, two places above Sampdoria on goal difference but with a game in hand.
(Reporting by Brian Homewood; Editing by Ken Ferris and Ian Chadband)
