Cabrera fired home from close range in the 63rd minute to guarantee the three points for the Sardinians at Trieste's Stadio Nereo Rocco.
The away side had taken a 27th-minute lead through Guglielmo Stendardo, but Cagliari hit back less than 60 seconds later through Radja Nainggolan before their comeback was completed by Cabrera.
The Rossoblu are still waiting for permission to return from exile and have to play their home matches over 1,000 kilometres north east of Sardinia near the Slovenian border.
Yet, this did not seem to affect them in the opening stages as first Nainggolan's effort from just outside the area forced a fine save from Atalanta goalkeeper Andrea Consigli, while Marco Sau can count himself unlucky after seeing his shot miss the target by inches having jinked his way into a good position.
Consigli had to be at his best once more to keep out Daniele Conti's curling free kick from the edge of the box.
Against the run of play, the visitors broke the deadlock when Stendardo powerfully headed home from Luca Cigarini's corner.
That lead would not last long, however, with the hosts responding immediately after falling behind when Albin Ekdal's cross found Sau, who teed up Nainggolan to slot past Consigli from close range.
The Atalanta shot-stopper kept his side level with another fine diving save to deny Chilean forward Mauricio Pinilla, while Sau once again missed the target with his effort.
Despite being on the back foot for the majority of the first half, Atalnata nearly re-took the lead just after the interval as Carlos Carmona's effort from distance was well saved by Cagliari keeper Michael Agazzi.
Consigli once more kept the scores level with a tremendous one-on-one save to deny Sau, who had burst into the area before being denied when clean through on goal.
However, it did not matter as Cabrera gave Ivo Pulga's side a deserved lead in the 63rd minute after converting from Nicola Murru's cross from the by-line.
La Dea nearly equalised mere minutes later but Marko Livaja's header struck the post.
Atalanta kept pushing resolutely for an equaliser but Livaja wastefully shot over from close range before the Croatian striker was once more denied by the woodwork from his effort inside the area.
Cagliari responded in kind, with Victor Ibarbo going past Atalanta defenders Gugliermo Stendardo and Mario Yepes before his strike was deflected by Consigli onto the post.