Naruto is certainly no stranger to combat, but over the course of the original series and Naruto: Shippuden, words were his preferred method of settling ideological differences -- so much so that fans have affectionately dubbed his attempts at dialogue with villains "Talk no Jutsu." However, while Naruto had a lot of success sweet-talking foes, when it came to reforming Sai, he didn't even have to speak to him.
Sai was first introduced as a replacement for Sasuke. He, along with Yamato, Sakura and Naruto, made up the new Team 7, which was responsible for capturing the supposed Akatsuki spy that had infiltrated Orochimaru's organization and bringing them back to the Hidden Leaf. It was his first mission with Team 7, and his already disastrous first encounter with Naruto had soured team relations. Unfortunately, he made matters worse by the derogatory tone in which he referred to Sasuke.
To Sai, Sasuke was just a traitor to the Hidden Leaf and the target of his secret mission sanctioned by the Foundation. The extent to which Naruto and Sakura went to defend Sasuke puzzled him, but also piqued his interest. After all the training he'd gone through to suppress his emotions, he couldn't understand how the two felt so strongly about their former comrade. Nevertheless, to keep the peace, Sai agreed not to insult Sasuke anymore in their presence.
The new Team 7 didn't instantly mesh, but they slowly became acquainted with each other. As they journeyed together, Sai revealed more about himself; he loved to paint and was preparing a picture book for his mysterious older brother, who he obviously held in high esteem. Nonetheless, he still made no effort to be friendly toward either Naruto or Sakura, so their decision to accept him for Sasuke's sake confused Sai all the more.
Eventually, Sai abandoned the rest of Team 7 and left with Orochimaru to complete his mission. In his haste, he left behind his sketchbook, which Sakura and Naruto investigated. In it, they found the story of Sai and his older brother. It was almost complete save for an empty page at the end, which was obviously for Sai's brother, and an unfinished drawing of Sai, which had no face. When confronted with his book, Sai revealed it was unfinished because he didn't remember his brother's face or what it was like to smile genuinely, but that wasn't his only disturbing revelation.
Sai disclosed that he'd met Sasuke during his and Team 7's time apart. According to him, Sasuke didn't acknowledge Naruto as a brother or comrade, and in fact advised Naruto to give up on him. To Sai's surprise, Naruto refused to give up on his friend, calling what they had a powerful bond that he was willing to risk anything to protect, even if it meant losing his own life.
Sai continued to observe Naruto as they searched for Sasuke in Orochimaru's hideout, becoming even more intrigued by him. The fervor with which Naruto continued to hunt for Sasuke despite his exhaustion amazed him. Naruto's spirit reminded Sai of his older brother, and his memories of working alongside him slowly began to return. When they stopped to rest, Sai took out his picture book and immediately began drawing; he had remembered his brother's face.
From then on, Sai's attitude toward Naruto and his mission changed completely. When he had an opportunity to assassinate Sasuke, he passed on it, instead attempting to capture the rogue ninja at great personal risk. Sai wanted to honor Naruto's attempt at redeeming his best friend, even if it meant going against the Foundation's orders. Their mission to retrieve Sasuke ended unsuccessfully, but the time he spent in Naruto's company forever changed his worldview.
Sai chose to further integrate into the Leaf Village. Despite the Foundation's principles of emotion suppression, he elected to seek out bonds among his ninja contemporaries. He studied books on how to be a better friend and formed relationships with some other members of the Konoha 11. His transformation was triggered not because his friends had advised him to change, but simply from observing how Naruto behaved and the steadfastness with which he clung to his bonds with friends. It made Sai remember that these bonds were well worth having.