The midfielder first rifled home Cenk Tosun's pass to level the score two minutes from time and then sent a glancing header past Sweden goalkeeper Robin Olsen, sealing the comeback win to the delight of the thousands of Turkish fans in the crowd.
Turkey, who suffered a 2-1 home defeat by Russia in their opening League B Group 2 game, started well and had lots of possession but the final ball into the box often lacked quality.
Sweden had taken the lead in their opening Nations League match when Isaac Kiese-Thelin scored 10 minutes before the break, flicking Viktor Claesson's cushioned header against the post before slotting home the rebound.
Claesson got on the scoresheet himself four minutes into the second half, drifting in from the left before releasing a rocket of a shot from distance that flew into the net.
Hakan Calhanoglu quickly threw Turkey a lifeline, pulling a goal back two minutes later with a deflected shot that wrong-footed Olsen.
The Swedes, whose defensive organisation helped them reach the quarter-finals at this year's FIFA World Cup in Russia where they were defeated by England, had several chances to increase their lead, but Akbaba had the final say to secure the points.
