In the evening's other tie, Xherdan Shaqiri, another former Bayern Munich player, scored his first goal for Inter Milan to set them on the way to a 2-0 win over nine-man Sampdoria after Mauro Icardi had missed a first-half penalty.
Gomez, beset by injuries and a loss of form since joining Fiorentina in 2013, finally enjoyed some luck when his shot from the edge of the area deflected off a defender to wrongfoot Vlada Avramov in the sixth minute.
Fiorentina doubled their lead through a Juan Cuadrado penalty in the 12th minute after the referee ruled that Guglielmo Stendardo had fouled Gomez as he was about to shoot, although the Atalanta player appeared to play the ball.
Atalanta protested furiously as Italian football was hit by a second refereeing storm in two days after AS Roma beat Empoli with a hugely controversial penalty in extra-time in another tie on Tuesday.
Atlanta coach Stefano Colantuono was sent off for dissent and later complained about the competition format, where the previous season's top eight Serie A clubs have a bye to the fifth round and are also given home advantage.
"There is no sense in it existing in this form, it is always the same teams who go a long way except with very rare exceptions," he told reporters.
Gomez, his confidence returning, then broke clear of the Atalanta defence to fire past Avramov before the half hour to seal the win and a quarter-final tie against AS Roma.
Rolando Bianchi scored Atalanta's consolation five minutes before halftime.
“Goals help a forwards, it's an advantage for him and the team and so will help him to have more confidence in himself," said Fiorentina coach Vincenzo Montella.
Sampdoria had Nenad Krsticic sent off after 11 minutes for tripping Shaqiri but held out for over an hour, with Icardi seeing a 37th minute penalty saved by fellow Argentine Sergio Romero.
Shaqiri broke through in the 71st minute with a powerful finish from Lukas Podolski's pass, then the visitors had Pawel Wszolek dismissed for a second bookable offence.
Icardi made amends for his earlier miss by pouncing on a defensive mistake to add the second for Inter.
(Writing by Brian Homewood in Berne; Editing by Toby Davis and Justin Palmer)
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