The Turkey international converted from close range 15 minutes from time to ensure last term's beaten UEFA Champions League finalists claimed their opening points in this season's competition.
Juve had not conceded a competitive goal under Allegri, who was appointed in July, prior to the Group A clash and that record looked set to remain intact as a result of a scrappy affair.
Atleti boast a fearsome recent home record in Europe - avoiding defeat at the Vicente Calderon in last season's competition - and the two sides cancelled each other out for much of a fixture that saw plenty of disruptions and stoppages due to fouls.
Mario Mandzukic's tame strike 25 minutes in was the only effort on target during the first half, while the in-form Carlos Tevez was kept quiet by the hosts, who were without injured captain Gabi.
However, when the sides seemed on course for a draw, Turan got on the end of a Juanfran cross to nudge home the winner and leave Allegri cutting a frustrated figure on the sideline.
Amid a fast-paced opening that saw possession change hands regularly, Leonardo Bonucci picked up a yellow card for tangling with Mandzukic after a reckless jump.
Atletico then had half-hearted penalty appeals turned down when Mandzukic and Turan felt Arturo Vidal handled in the area.
The Croatia international stung the palms of Gianluigi Buffon from distance after 24 minutes, in an opening half-hour devoid of clear-cut chances.
Paul Pogba's curled effort from outside the area veered wide of Miguel Angel Moya's left-hand post as the latter stages of the first half became even more scrappy.
With fouls aplenty from both sides, the game remained fractured and there was no sign of a goal prior to the interval.
Juventus dominated the opening exchanges after the break but, as in the first half, failed to cause Atletico too many problems.
Another Turan delivery led to shouts for a penalty as it deflected off Stephan Lichtsteiner and struck Martin Caceres on the arm - although referee Felix Brych remained unmoved.
However, Turan was to come up with the goods for Simeone as he timed his run into the area to get on the end of Juanfran's cross after Mandzukic had failed to connect with Juanfran's delivery.
The winger - involved in much of the limited attacking play from the hosts - turned home the winner to hand the hosts maximum points on Simeone's return to the dugout following his UEFA ban.
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