The visitors produced the better chances in Naples but couldn't find a way past the Colombian goalkeeper, who reacted sharply to block dangerous efforts from Matteo Pessina and Luis Muriel either side of the break.
Napoli coach Gennaro Gattuso opted for a new 3-4-3 formation, but his side struggled to carve out opportunities in the unfamiliar system and could only muster two shots on target to Atalanta's six.
"Today the collective did well," Ospina told Rai Sport.
"We knew it would be a very difficult game against a team that plays very well. We did our job, always thinking about winning."
The two sides face off in the second leg in Bergamo on Feb. 10 and Juventus or Internazionale await in the final, with the Turin club in charge of that tie thanks to a 2-1 first-leg win at San Siro on Tuesday.
Gattuso complained after Sunday's 2-0 Serie A win over Parma of being "slaughtered" by the media and expressed his disappointment at the speculation over his future that followed back-to-back defeats to Juventus and Hellas Verona in January.
But his side failed to impress at the Stadio Diego Armando Maradona as Atalanta created the better chances.
Ospina rushed out to block a Pessina effort with his leg before Rafael Toloi cleverly found space in the box only to poke his finish wide.
The Colombian rushed off his line to smother Muriel's finish early in the second half and the goalkeeper was alert again to hold a glancing header from his compatriot Duvan Zapata.
Atalanta defender Cristian Romero picked up a booking that rules him out of the second leg through suspension, while Napoli midfielder Diego Demme had to be taken off on a stretcher after the ball struck him in the face.