The Istanbul team are bothered after three injuries as Brazilian midfielder Luiz Gustavo, Ecuadoran forward Enner Valencia and Turkish goalkeeper Altay Bayindir sustained injuries each to leave the match in the second half.
Bayindir burst into tears while leaving the pitch in the 69th minute as he collided with Konyaspor's Serdar Gurler after a cross. Berke Ozer checked in.
Home team Konyaspor took an early lead as Soner Dikmen made a long range effort to beat Fenerbahce goalie Bayindir in the second minute.
Konyaspor made it 2-0 in the minute 11 as Abdulkerim Bardakci scored a header in the far post after Guilherme's cross from edge of the penalty box.
The scorer was free at the far post.
In the 55th minute of the match, Valencia's bicycle kick hit the crossbar.
Fenerbahce narrowed the gap to one after Turkish midfielder Irfan Kahveci curled from freekick, a classy long range effort in the 84th minute.
After Valencia's injury in the minute 80, Fenerbahce had to carry on with 10 men as the visitors previously made all the five substitutes.
Konyaspor secured a 2-1 victory to reach the fourth position in standings.
They collected 20 points in 11 matches.
Sixth-place Fenerbahce have 19 points.
Trabzonspor - Caykur Rizespor: 2-1
Kasimpasa - Oznur Kablo Yeni Malatyaspor: 2-0
Altay - Demir Grup Sivasspor: 1-1
Atakas Hatayspor - Besiktas: 1-0
Ittifak Holding Konyaspor - Fenerbahce: 2-1