Italy defender Bonucci smashed home almost identical spot-kicks in each half at the Stadio Olimpico in Rome to move Juve sixth, level on 21 points with fifth-placed Lazio.
Massimiliano Allegri's side are now four points from the Champions League positions, with Atalanta occupying fourth following their 5-2 thumping of Spezia earlier on Saturday.
It was a dogged, solid away performance typical of Allegri's best sides and a good sign ahead of Tuesday's trip to Stamford Bridge to face Premier League leaders Chelsea.
"It was a really important win, it's always hard to play matches after the international break but we know that with this approach and spirit of sacrifice that results depend on us," said Bonucci after his first Serie A double.
"When you're Juve and played like we did at the start of the season it's right that you get criticised, but that should encourage us to improve. It's down to us to maintain the same spirit from tonight."
Bonucci opened the scoring with his second league goal of the season with 22 minutes gone, rifling a shot into the top corner past Pepe Reina after Danilo Cataldi was punished for a silly challenge on Alvaro Morata following a VAR check.
Bonucci could have doubled the away side's lead almost straight after the break when he beat Reina to the ball after a Manuel Locatelli corner but could not find the net.
Bonucci sinks Lazio
He made no mistake from the spot with seven minutes remaining however, lashing home into the same corner after Federico Chiesa was chopped down in the box by Reina - the Lazio goalkeeper having come charging out of his box to try to stop the rampaging Italy forward.
Moise Kean was unlucky not to make the scoreline even more emphatic late on after Reina twice in two minutes had to keep out decent efforts from the substitute.
The win for Juve was their first by more than one goal in Serie A this season, and they stay some way off pace-setters Napoli and AC Milan, the latter of whom will go top with a win or draw at Fiorentina later on Saturday.
Gian Piero Gasperini's Atalanta are level on 25 points with Inter Milan, who sit third and host league leaders Napoli on Sunday evening, after a demolishing Spezia in Bergamo.
A Mario Pasalic brace and Duvan Zapata's penalty, which the Colombian forward scored at the second attempt after Spezia players encroached during the first saved spot-kick, ensured a comfortable first-half lead for Atalanta after M'Bala Nzola had opened the scoring for the away side.
Late goals from Luis Muriel and Ruslan Malinovskyi then wrapped up the points and consolidated Atalanta's place in the Champions League positions as they warmed up nicely for Tuesday's trip to Young Boys.
Spezia sit in 16th, just two points above the relegation zone and regional rivals Genoa and Sampdoria, after their eighth defeat in 13 games so far this season.
Standout matches today