Serie A: Atalanta 0 Juventus 3
The reigning champions made it five wins from five in Serie A this season as a double from Tevez and a first Juve goal for substitute Alvaro Morata made the points safe.
It was not all plain sailing for Allegri's side, however, and in the end they were thankful to Gianluigi Buffon for keeping out German Denis' second-half penalty, which would have levelled things up at 1-1.
After finally getting to grips with the midfield battle, the hosts edged ahead 10 minutes before the break when Tevez – fresh from being rested in Wednesday's win over Cesena – tapped in.
Giorgio Chiellini hauled down Salvatore Molina to give Atalanta the chance to restore parity from the penalty spot in the 59th minute, but Buffon denied Denis and Argentinian Tevez then punished his compatriot's half-hearted effort less than 60 seconds later.
From there, Atalanta never looked like mounting the comeback required and with seven minutes left Morata claimed his first goal for his new club since his close-season move from Real Madrid, nodding home to complete a comfortable win.
Allegri had been keen to ensure that his players did not take Atalanta lightly in the build-up and the hosts certainly posed an early threat in Bergamo.
On-loan Richmond Boakye, facing his parent club, did well to bring down a lofted ball on the edge of the area and Carlos Carmona lashed a half-volley just wide following the Ghanaian's lay-off.
Chances were infrequent during a lacklustre opening, but things did begin to liven up as the half-hour mark approached.
First, former Juve winger Marcelo Estigarribia shot across the face of goal in the 25th minute, before Marco Sportiello denied Tevez at his near post a few moments later.
Juve eventually found the breakthrough in the 35th minute, and it came as no surprise to see Tevez get the goal.
Fernando Llorente showed good awareness to play Stephan Lichtsteiner into the area and, after the Swiss defender's squared pass had squeezed through a group of players, Tevez was on hand to tap in.
Juve should have added a second just before the break, as Llorente nodded Paul Pogba's sliced volley agonisingly over.
After a slow start to the second half, the match once again burst into life just prior to the hour mark.
Chiellini brought down Atalanta substitute Molina to concede a penalty in the 59th minute, but Denis saw his tame effort pushed away by Buffon and Juventus punished them almost immediately.
Tevez received Patrice Evra's cut-back on the edge of the area and, although Sportiello got a hand to his effort, the Argentinian's attempt squeezed inside the post.
Molina should have pulled a goal back for Atalanta in the 78th minute when he produced a clever first touch to cut inside Chiellini, only to blaze his shot well over from close range.
Juve invited pressure on to themselves in the closing stages as they looked to catch their hosts on the counter.
And their tactics paid off with time running out as Morata, on as a substitute for fellow Spaniard Llorente, nodded in Roberto Pereyra's chipped cross in the 83rd minute.