Juve's win took them four points clear of rivals Napoli, who visit Fiorentina on Sunday, and ensured they will finish the weekend top of the table with three matches to play.
Juve failed to impress as they played with a one-man advantage for more than one hour but did take the lead through Douglas Costa on 13 minutes when he drilled a low shot into the bottom corner.
Cuadrado's equaliser from a tight angle was slightly deflected by Milan Skriniar on 87 minutes before Higuain headed in the victor off substitute Paulo Dybala's free-kick two minutes later.
Inter looked as if they would pull off an unlikely victory as the hosts led until three minutes from the whistle despite playing more than 70 minutes a man down after Matias Vecino was sent off.
The supporters who had gathered chanted support for their team, encouraging them to fight to the end in order to win their seventh-consecutive league title. "I think they came to encourage us, they did a good thing", revealed veteran defender Andrea Barzagli to Sky Italia. "We know that to win the championship we have to win all of our games". It is very hard to win at San Siro and the lads put in an extraordinary performance.
Minutes later, the sending off of midfielder Matías Vecino didn't help the hosts as the Uruguyan went in for a strong challenge that caught forward Mario Mandzukic on the ankle. Gonzalo Higuain headed the victor in the 89th minute.
Alli, Kane Score To Help Spurs Beat Watford
I am sure that, first of all, he will perform for us and no doubt he will arrive in very good condition for the World Cup. Spurs are also five points ahead of Chelsea having both played 35 times with the Blues set to visit Anfield on Sunday.
Juve coach Massimiliano Allegri was sent off for entering the pitch after Higuain's goal.
Eusebio Di Francesco's side is three points ahead of Lazio which plays Torino Sunday with Inter staying a further point back in fifth.
The Giallorossi will hope to repeat their heroics against Barcelona in the last round by overturning a 5-2 semi-final first-leg deficit against Jurgen Klopp's men on Wednesday and found some form on home soil.
Inglese headed in a late consolation goal for Chievo, but they remain two points above the drop zone.
Patrik Schick and Dzeko scored in a first half dominated by Roma, who had Juan Jesus dismissed for a last-man foul on Roberto Inglese on 56 minutes. "Us with the voices, you with the heart, let's win it!", read a banner displayed by the Drughi Ultras group.