Fagioli sealed the win for the visitors in the 73rd minute when his curled shot from the edge of the box bounced in from the right post, after an assist by Samuel Iling-Junior who came on to the pitch one minute earlier.
"The important thing was to win because we were coming from a bad defeat against Benfica," Juve manager Massimiliano Allegri said after they lost 4-3 in Lisbon on Tuesday to exit the Champions League at the group stage.
"In football, as in life, you always fall down and get back up. The important thing is to have courage -- everything can't always go right or wrong. [But] you have to be reckless and have a healthy madness when things go wrong.
"We managed to switch from the Champions League to the championship, I just have to thank [the boys] for what they did.
"Responses are crucial. It was since March last year that we hadn't won three games in a row, [and] tonight was definitely not easy."
After a dull first half with no threat to either goal, Juventus midfielder Adrien Rabiot came close to nodding in a corner from Filip Kostic just before halftime but Lecce goalkeeper Wladimiro Falcone dived full-stretch to deny him.
Lecce came close to levelling shortly before the final whistle but midfielder Morten Hjulmand hit the left post with his fine strike.
Juventus provisionally moved two places up to sixth in the standings on 22 points, one ahead of Inter -- who host Sampdoria later on Saturday -- and Udinese, who travel to bottom-placed Cremonese on Sunday.
Juventus, who will focus on sealing a place in the Europa League when they host Paris Saint-Germain on Wednesday after their Champions League exit, host Inter in Serie A next Sunday.
Lecce, who remain 17th on eight points from 12 games, travel to Udinese on Friday.