Gonzalo Higuain came off the bench to score the winner in an intense Derby d’Italia clash as former Juve boss Antonio Conte suffered a first league defeat in charge of Inter.
Sarri told juventus.com: “Overtaking (Inter) at this time has little meaning. What matters is the performance and character shown against a strong team, which had always won up to this moment.
“We entered the game with a good approach, and we deserved to win.”
Paulo Dybala’s opener was cancelled out by a Lautaro Martinez penalty before Higuain’s late decider.
Sarri acknowledged the defending champions need to work on dominating possession more following the narrow win.
He added: “From this point of view, we have already taken important steps.
“What we need to work on is dominating the match, having territorial supremacy, pressing better and more, taking away any attacking initiative from the opposition so that we can spend our whole time in the opposition half.
“We have room for improvement, we are not 100 per cent yet, but the team is strong so we can get the results.”
Conte – who kick-started Juve’s run of eight straight titles by delivering the first three – came out second best against his former side.
The fellow former Chelsea boss told inter.it: “It was a match against a very, very strong team and I’m sorry about the loss.
“We could have done better with our finishing and the final ball but the team did what it could. Looking at the statistics, it was a closer game than perhaps shows on paper.”
The home side lost Stefano Sensi to injury after 33 minutes and Conte believes his absence hindered Inter’s performance.
He added: “We paid for the Sensi injury, losing our way a bit.
“At the start of the second half, we didn’t do well but then we got back into the game and either side could have scored.
“There were chances at both ends but they scored, using everything in their arsenal and making their experience count.
“I don’t have anything to say to the guys because we still have a long way to go in terms of how we manage and understand moments in a match.
“Congratulations to them, we’ll roll up our sleeves and keep working hard to move forward on our journey.
“There are no shortcuts in this regard, they’re at a different level and we’re looking to improve so that we can close the gap.”
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Under-fire AC Milan boss Marco Giampaolo admitted the pressure got to his players as they crashed to a 3-1 home defeat by Fiorentina.
The Rossoneri slumped to a fourth loss in six matches and had Mateo Musacchio sent off for a nasty lunge on Franck Ribery.
The defeat leaves a huge question mark over the future of Giampaolo, with reports in Italy claiming he would be sacked were they to lose on Sunday night.
Goals from Erick Pulgar, Gaetano Castrovilli and Ribery did the damage for Fiorentina, who also missed a penalty through Federico Chiesa, with Rafael Leao pulling one back.
“I believe in my ideas, but this Milan team looked like it turned up without ever having a training session together,” Giampaolo said on Sky Sport Italia.
“The team played with great character just three days ago (against Torino) and I saw moments of the football I like. Tonight, it played badly on an individual level, with no organisation or sense of collective responsibility.
“There are some players who are young and can try individual moves, but they can only get you so far. When things aren’t going well, you must have the strength to cling to your identity as a team.
“If we try to resolve things individually, we risk drowning in difficulties and that is what happened this evening.
“I take responsibility, of course, but I go forward because I believe in my ideas. The thing that annoyed me was the team looked like it turned up to San Siro without ever having a training session together. You can lose, but not like that.
“Up until three days ago, I liked the performance and saw good responses against Torino.
“Tonight, the pressure got to them, they felt the absolute need to win and I told them to play with all the right ingredients to win, not waiting for events to happen. We had to create the events ourselves.
“Instead, the pressure weighed on us, we were tense, not reacting quickly enough and evidently feeling the pressure.”
For Fiorentina it was a second win in five days having picked up their first victory of the season against Sampdoria in midweek.
Viola head coach Vincenzo Montella was full of praise for his side and singled out Ribery.
“It’s a great achievement to beat AC Milan at San Siro,” Montella told the club website.
“It’s never easy to play here with the fans whistling at you. We eased off a touch at 3-0 but overall I’m pleased.
“Ribery deserved the standing ovation. Once again he made the difference out there – he always does.”
Reaction from Eugenio Corini after Mario Balotelli failed to inspire Brescia to victory against a Cristiano Ronaldo-less Juventus in the Italian Serie A on Tuesday.
Balotelli, 29, was kept quiet on debut for his hometown club when the Serie A champions won 2-1.
Here is the reaction of his head coach: