Mario Yepes's 82nd-minute header completed the fightback for the defending champions who trailed to goals from Uruguayans Guillermo Giacomazzi and Carlos Grossmuller and a Massimo Oddo penalty after a shambolic first-half display.
Argentine Erik Lamela, 19, scored seven minutes into his Serie A debut in AS Roma's 1-0 win over Palermo.
Inter Milan won for the only the second time in seven Serie A outings this season when a Thiago Motta header gave them a far from convincing 1-0 win over Chievo.
Udinese, unbeaten after seven games, lead with 15 points while Lazio are second, one point behind, after a 2-0 win at Bologna in the late match thanks to an own goal from Robert Acquafresca and a Senad Lulic strike.
Juventus, also unbeaten after being held 2-2 by Genoa on Saturday, dropped to third with 13 and Cagliari are fourth with 12 after a goalless draw at home to Napoli.
Napoli, Roma and Milan are level on 11 points.
"I'm happy because Milan are finally where we deserve to be," Boateng, whose side started the weekend a modest 13th, told Sky Sport Italia.
"This isn't a reply to my critics because I look to the future and not to the past."
Milan coach Massimiliano Allegri added: "A comeback win like this is very important. We got ourselves into a bit of a mess and we did little in the first half, we were too slow, although I had faith we could fight back. I'm happy for Boateng."
Giacomazzi headed Lecce in front after four minutes and former Milan defender Oddo converted a harshly-awarded penalty on the half hour following a foul by Christian Abbiati on Daniele Corvia.
On a bumpy pitch, a mistake by Mark van Bommel set up another Lecce counter-attack seven minutes later which Grossmuller finished off by slotting the ball past Abbiati.
The game changed completely when the ineffective Robinho was replaced by the heavily-tattooed Boateng at half-time.
The Ghana international fired in from close range within four minutes and scored with an exquisite first-time strike from 25 metres following Antonio Cassano's pass six minutes later.
He levelled with an opportunist goal in the 63rd minute when a rebound from Ignazio Abate's shot landed at his feet, although Yepes was clearly offside at time.
Antonio Nocerino hit the bar and Zlatan Ibrahimovic just failed to turn in a Cassano pass for Milan before Cassano's delicate cross was headed in by Yepes with eight minutes left.
Inter dominated the first half against Chievo at the San Siro with midfielder Wesley Sneijder looking his old self and went ahead when Thiago Motta rose above the defence to score with a powerful header in the 34th minute.
Maicon hit the post for Inter in the second half but they were never enti
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