Gattuso guided Napoli to their first title since 2014 courtesy of a 4-2 penalty shoot-out win against Juve in Rome on Wednesday.
Penalties were needed in the final following a scoreless draw behind closed doors in the Italian capital midweek.
De Laurentiis lauded the transformation overseen by former Milan midfielder and coach Gattuso, who replaced Carlo Ancelotti in December amid the club's struggles.
"It was in the air. Since Rino Gattuso arrived, many things changed here and everyone came together around him, around the club, around the idea of Napoli," De Laurentiis told RAI Sport.
"After all, at the end of the day, Napoli are the only club capable of contrasting Juventus for this title.
"We beat them in the Coppa Italia in 2012, we beat them in the Supercoppa, and so far we haven't been able to beat them only in Serie A."
Napoli, who defeated Inter en route to the Coppa Italia decider, will restart their Serie A campaign against Hellas Verona on Tuesday following the coronavirus pandemic.
Gattuso's Napoli are sixth in the standings through 26 games, six points behind Roma and nine adrift of fourth-placed Atalanta.
Connery – best known for his role as fictional British secret agent James Bond in the iconic film series between 1962 and 1983 – died aged 90 on Saturday.
The Scotsman was also a Rangers supporter, while previously serving as a member of the board at Ibrox.
Gattuso – a Champions League and Serie A winner with Milan and World Cup champion with Italy – spent a year playing for Rangers before returning to his homeland via Salernitana in 1998.
Asked about Connery following Napoli's 2-0 loss to Sassuolo on Sunday, Gattuso told reporters: "I only met him once, the president back then was David Murray, he was a member of the board.
"It's true, we spoke on an evening and he absolutely didn't want [me to leave]. What can I say?
"A very big hug to his entire family, the people who represent him, surely I've got beautiful memories of him, very fascinating, great man, he made history in cinema."
Napoli were upstaged by high-flying Sassuolo, who stayed second and within two points of unbeaten leaders Milan.
Manuel Locatelli and Maxime Lopez scored in the second half to leave Gattuso's Napoli fifth and five points off the pace.
Locatelli converted a 59th-minute penalty to become the youngest Italian with two or more goals in Serie A this season, and the youngest player with a penalty goal in the competition this term.
Sassuolo have collected 14 points from their first six games in Serie A – a record for the club after their first six matchdays in a single top-flight season.
"We mustn't listen to people talking about the Scudetto," said Gattuso, whose Napoli edged Real Sociedad in Europa League action on Thursday. "We'll try to get into the top four. It's a very strong league this season, every team is smart, they change their shape, and it is really difficult to find a way through.
"I see my Napoli team. We're not as sharp as we were before, but with a game every two, three days, that's only to be expected."
