Inter had briefly led in Italy after Lukaku slotted home a 92nd-minute penalty to snatch the three points at Bologna, but Juve retook first place thanks to De Ligt, who smashed home the ony goal of the game in Turin with 20 minutes remaining.
Maurizio Sarri's side stay a point ahead of Inter thanks to a hard-fought derby win that saw both sides exchange blows in a bruising contest.
"I am really happy, of course. It was a really important game, as Inter won also. We won 1-0 and it was my goal, so really special," De Ligt told DAZN.
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De Ligt has had a shaky start to life in Italy and could well have given away an early penalty when he touched the ball with his arm early in the first half, but was let off by referee Daniele Doveri and the VAR officials.
However the Netherlands international, who had already been denied by a superb Salvatore Sirigu stop just before the break, came up trumps in the 70th minute when he met substitute Gonzalo Higuain's pull back just after the Argentine had forced a corner with a sizzling volley.
"Obviously, when you come to a new club, it's always going to be hard. It's going well, the guys made me feel welcome and I think I'm improving every game," added De Ligt.
The win maintains Juve's unbeaten start to the season and keeps them on course for a ninth straight league title, while Inter have to content themselves with second despite Lukaku's match-winning display.
The Belgian became the first player since Ronaldo in 1997/98 to score at least nine goals in his first 11 Serie A matches as an Inter player after a 75th-minute tap-in which cancelled out Roberto Soriano's deflected opener and a stoppage time spot-kick winner.
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"I'm pleased to have had such a good start, but I am 26 years old and I need to grow every day, along with the whole team," Lukaku told Sky Sport.
Inter have been tipped as the principal challengers for Juve's Serie A title but had to be saved from a lacklustre display by their star summer signing, who arrived from Manchester United for 65 million euros ($73 million).
His form will be a big boost going into Tuesday's huge trip to Borussia Dortmund, which could make or break their Champions League campaign with both sides level on four points behind Group F leaders Barcelona.
Inter are also six points ahead of Roma, who moved third in Serie A on Saturday after their 2-1 win over wobbling Napoli.
Rising star Nicolo Zaniolo opened the scoring midway through the first half with his fourth goal in as many games in all competitions before Jordan Veretout sealed the points from the penalty spot 10 minutes after the break and stretched Roma's unbeaten league run to six games.
"I'm very pleased with my current form, I hope to continue like this and to score many more goals," Zaniolo told Sky Sport.
Arkadiusz Milik's fifth goal in four games closed the gap for Napoli, but the defeat sees them drop down to seventh, level on 18 points with Lazio and Cagliari and 11 behind Juve after winning just one of their last five in Serie A.
Paulo Fonseca's Roma meanwhile are on a fine recent run despite an injury crisis that kept a host of first team starters out of action over the last few weeks.
They are set to have further selection problems next week after Mert Cetin, who replaced suspended Federico Fazio, was sent off for two bookable offences deep in stoppage time.
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