Haris Medunjanin got forward a few times early on as the home side had the better of the possession in the opening 20 minutes, but he was unable to make the most of his chances.
Getafe started to see more of the ball as the half progressed though, and took the lead shortly before half time when Abdoul Karim Yoda struck the post, only for the rebound to strike Fabri and end up in the back of the net.
Sergio Escudero could have doubled their lead moments later after beating several defenders with a quick run, but his powerful shot from the edge of the box went just wide.
Mehdi Lacen had a chance to end any hopes of a comeback by the hosts with just over 10 minutes to play when he got the ball in space on the edge of the box, but his poor finish flew well high.
The miss proved costly as Deportivo pulled one back moments later when Toche sent in a ball for Postiga, who unleashed a powerful strike to get the home side back into contention, but they were unable to find an equaliser before the full-time whistle.
The win lifts Getafe up to ninth in the table, while Deportivo, who have just one win from their last seven league matches, sit 16th.