The victory elevated the Bergamo side to third, level on 66 points with Inter Milan but ahead on head-to-head record.
Gian Piero Gasperini’s side will be assured of a top four finish if they beat Sassuolo in their final game of the season next weekend.
“It’s a positive result,” Gasperini told Sky Italia.
“Now we need to reach the finish line, so much work has been done and now we must go for it against Sassuolo. We still need to give everything."
Juventus, who clinched an eighth successive Serie A title almost a month ago, announced this week that Allegri would leave the club at the end of the season.
The home crowd bid farewell to the coach after five trophy-laden years, while veteran defender Andrea Barzagli also took to the Allianz Stadium pitch for the final time.
Juventus's form has dipped dramatically since they secured the title and they have now gone four league matches without a win for the first time since 2012.
Cristiano Ronaldo squandered a chance early on by blasting over the bar from a good position and they were made to rue that miss when Ilicic tapped in Atalanta’s 100th goal of the season in all competitions after 33 minutes.
Barzagli, who will be retiring after more than eight years at the club, was given a guard of honour by his team mates before the game and that was followed by a standing ovation when he was substituted in the second half.
Juve drew level when Mandzukic slid in to meet Juan Cuadrado’s looping cross at the back post and struck the ball into the net through Pierluigi Gollini’s legs.
The hosts were reduced to 10 men in stoppage time when Federico Bernardeschi was sent off for a rash challenge.
(Reporting by Alasdair Mackenzie, editing by Pritha Sarkar)
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