Juve's Brazilian-born forward Amauri rose between two markers to head home a Diego freekick in the 72nd minute to break the deadlock after the visitors had struggled to pose a threat for much of the match.
The Turin side are third with 18 points from nine games, four behind Inter Milan and two shy of second-placed Sampdoria, after the leading duo beat Catania 2-1 and Bologna 4-1 respectively on Saturday.
Milan are three points further back in joint fourth after Nesta's headers in the 81st minute and in injury time wiped out Giampiero Pinzi's seventh-minute strike for the hosts.
They share fourth spot with Fiorentina, who suffered their first home defeat of the season by going down 1-0 to Napoli, and Palermo, who beat Udinese 1-0 thanks to Cesare Bovo's deflected late volley.
Juventus had plenty of problems with the defensive wall erected by their Tuscan opponents and centre back Giorgio Chiellini headed their only decent chance of the first half wide.
But they reaped the maximum reward for raising the tempo briefly in the middle of the second half and Amauri's goal sank Siena to the foot of the table, putting more pressure on coach Marco Giampaolo.
"It was fundamentally important for us to get back to winning," Juve coach Ciro Ferrara, who made a number of changes to the side who beat Maccabi Haifa 1-0 in the Champions League on Wednesday, told Rai television.
"We managed to do that even though we had problems. I think it bodes well for Wednesday's match (against Sampdoria)."
Milan left it late in Verona but they had laid siege to the Chievo goal in the second half and deserved to continue their revival from a shaky start after comeback wins over AS Roma and Real Madrid.
Goalkeeper Dida, guilty of a terrible blunder in Wednesday's 3-2 win in Madrid in the Champions League, was one of the heroes of the night with a pair of excellent saves.
Livorno moved off the bottom by beating AS Roma 1-0 away thanks to former Roma forward Francesco Tavano's 40th-minute strike to give new coach Serse Cosmi a win in his first game in charge despite having goalkeeper Alfonso De Lucia sent off in the 64th.
