The Monegasque finished 2.664 seconds ahead of McLaren's Oscar Piastri, who had seized the lead on lap one from teammate and title contender Lando Norris, as Ferrari made a bold strategy decision and reaped the rewards.
Norris ended up third after starting from pole position at Monza's super-fast circuit and is now 62 points behind Red Bull's Max Verstappen, who finished only sixth, with eight races remaining.
Leclerc succeeded by making just one pitstop, to McLaren's two, and nursing a set of hard tyres for 38 of the 53 laps as he hung on for victory.
"Mamma Mia, Mamma Mia," he screamed over the team radio as former Italy soccer player Alessandro Del Piero waved the chequered flag.
"It's an incredible feeling," Leclerc said before the podium celebrations and traditional track invasion by an army of fans swamping the pit straight with a tidal wave of red shirts and flags.
"I want to win Monza and Monaco every year and I have managed to do so. It is so, so special."
Leclerc, a winner at Monza in 2019 and at home in Monaco this year, was 11 seconds clear with seven laps to go and 8.3 ahead with five remaining as the crowd willed him on to his second victory of the season.
Piastri, who said the defeat hurt, had pitted for a second time on lap 39 after Norris did so at the end of lap 32.
"We considered a one-stop strategy the whole race but it was not possible with the amount of (tyre) graining I had," said Norris, who took a bonus point for the fastest lap.
"We are disappointed but Ferrari drove a better race."
McLaren are now eight points behind Red Bull, down from a previous 30.
They opened the door for Ferrari on the opening lap when Piastri went past Norris at the second chicane - an aggressive move that forced the Briton to back off and allowed Leclerc, who started fourth, to speed past.
Norris took back the place by pitting earlier than Leclerc, an undercut that kept him ahead when the Ferrari came in, but the strategy then unravelled and McLaren were left with questions to answer about team priorities and tactics.
"Oscar caught me by surprise as he got past," Norris told reporters.
"I don't know what I could have done differently. If I brake a metre later, I probably would have crashed."
Carlos Sainz was fourth for Ferrari, overtaken by both McLarens in the final stages, with seven-times world champion and future Ferrari driver Lewis Hamilton fifth for Mercedes.
George Russell was seventh for Mercedes after starting third but running off the track and down as escape road at the start, damaging his car.
Red Bull's Sergio Perez finished eighth, continuing a sequence of disappointment, and Alex Albon secured precious points in ninth for Williams who had Argentine Franco Colapintro finish 12th on his debut as replacement for Logan Sargeant.
Kevin Magnussen was 10th for Haas despite a 10-second penalty for causing a collision that triggered an automatic ban for the next race in Baku.
