The three-time Asian champions, who also missed a penalty just after the break, became the second nation after five-times champions Brazil to qualify for the finals by securing a fifth World Cup appearance at the 32-team tournament.
Iran's president Hassan Rouhani congratulated the nation and the players on clinching a place at the finals. "Your victory made all of us happy... Guys well done for the qualification for the 2018 FIFA World Cup in Russia," he tweeted.
Thousands of Iranians spilled onto the streets after the match with impromptu street parties in Tehran and across the country as people danced, waved flags and honked car horns.
"People are shouting 'Iran, Iran'... It is like a carnival … We are so happy and proud of our team," said Iran fan Reza Ghanbari in northern Tehran's Tajrish Square.
The win means Portuguese coach Carlos Queiroz's Iran team will finish in the top two places in Group A of Asia's final round of qualifying. Uzbekistan are eight points adrift of the leaders in third place and have only two games remaining.
The top two teams in each of Asia's two qualifying groups advance automatically to the finals while the teams in third place go through a series of playoffs.
Azmoun opened the scoring in the 23rd minute when Alireza Jahanbaksh's through ball found the FC Rostov striker with time and space behind the Uzbek defence to roll the ball under goalkeeper Aleksandr Lobanov and into the net.
Captain Masoud Shojaei squandered the chance to double Iran's advantage five minutes into the second half when he fired a penalty over the bar after Taremi had been bundled over in the box by Akmal Shorakhmedov.
But Taremi calmed Iran's nerves when he added the second goal late on by curling a low shot into the corner of Lobanov's goal to spark the celebrations amongst the home supporters.
(Writing by Michael Church in Hong Kong; Additional reporting by Parisa Hafezi in Ankara; Editing by Ken Ferris)
