The League One side took the lead through Jordan Obita, before goals from Victor Anichebe and Phil Jagielka gave the visitors a hard-fought advantage. However, that was not the end of the story as Smith powered in a corner to keep his side's dream alive.
Everton dominating much of the early going. However, Moyes' charges struggled to create too much in front of goal, with Leighton Baines' deflected free kick their best chance of the opening stages.
Everton was soon made to pay for its profligacy, as it quickly fell behind just minutes after that Baines opportunity. Lee Croft began a rapid counter and found the perfect cross which Obita tapped home from close range, with Tim Howard stranded.
The home side's lead lasted less than 10 minutes however, as Everton turned the screw before finding the leveller through Anichebe, the Nigerian powering home after Nikica Jelavic had flicked a long ball from Jagielka into his path.
Oldham was not deterred by Everton's response, and it almost re-established its advantage seconds after the equalizer, Obita again at the forefront as he saw a curling effort strike the post with Howard beaten.
Tony Philliskirk will have been pleased to see his side go into the interval with the tie all-square, however that satisfaction proved short-lived as Jagielka rose to meet a corner from substitute Kevin Mirallas and give the visitors the lead.
Oldham remained a threat despite Everton's relative dominance, and came close to restoring parity, when Smith, the hero of the win over Liverpool, forced an excellent stop from Howard with a fierce low drive.
However the Latics continued to struggle with the quality of the Premier League side from a defensive standpoint, and Jelavic almost ended the game as a contest when he fired over following a good through ball from Darron Gibson.
Still the hosts kept pushing forward, and they must have been cursing their luck as Lee Barnard witnessed his close-range strike deflect agonizingly wide of the post before Howard was called into action again to deny Robbie Simpson.
Undeterred, Oldham fought desperately to try and provide the sting in the tail, and it was Smith who was once again the headline maker, leaping highest to meet a Jonathan Grounds corner and spark jubilant celebrations from the home crowd.
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