It took the visitors just four minutes to break the deadlock, Antonio Sanabria heading home Alberto Lora's cross to convert the first chance of the afternoon.
Sanabria doubled his tally with a near-identical header four minutes later, with Lora serving as the provider again as Sporting Gijon edged further in front.
Deportivo pulled one back in the 16th minute through Juanfran, who headed home from a tight angle after Lucas Perez swung a corner into the penalty area.
The hosts were level just before the half-hour mark, and their equaliser came from another corner, Luis Alberto heading Pedro Mosquera's in-swinging cross over the line.
The parity lasted for just six minutes as the visitors went back in front in the 34th minute following a swift break, which ended with Alex Menendez smashing Alen Halilovic's low pass into the bottom corner.
There was more end-to-end action in the second period, but tighter defending at both ends of the park meant that was the end of the goal rush.
Sporting Gijon withstood an onslaught through five minutes of stoppage time to claim their first win of the campaign, moving up to ninth place in the table in the process.