Ismail Kartal's men came into the weekend joint top with Besiktas, with whom they shared an identical league record - same points, same goal difference, same number of wins and the same number of goals scored.
However, Fenerbahce went behind twice on Sunday and were lucky to escape with a point - Moussa Sow netting a stoppage-time penalty to salvage a draw.
Erkan Zengin opened scoring for the visitors when he found space on the left-hand side of penalty area and drove a low effort beyond Volkan Demirel before Pierre Webo levelled matters by poking home after the ball rebounded off the crossbar.
Emre received his marching orders for a second bookable offence five minutes from time and Rogelio Funes Mori appeared to have sealed the win when he rounded the keeper to score soon after.
However, the referee pointed to the spot following an alleged handball, with Sow stepping up to take full advantage and save further embarrassment for Fenerbahce.
Fenerbahce are now two points adrift of Besiktas, who were 2-1 winners at Karabukspor on Monday.
Demba Ba scored twice in two second-half minutes to put the league leaders in control - lifting over goalkeeper Boy Waterman before heading home the second.
Musa Cagiran pulled a goal back seven minutes before the end, but Besiktas held out to claim their place as outright leaders in the Turkish top flight, although they did see substitute Mustafa Pektemek dismissed for a wild lunge in stoppage time.
Galatasaray sacked coach Cesare Prandelli on Friday, but bounced back with a 1-0 win at Gaziantepspor a day later to go second.
Gala then named a new boss on Monday, with Turkey assistant Hamza Hamzaoglu taking the reins.
A change of coach certainly seems to have worked for Trabzonspor, who are on a three-match winning run in all competitions since appointing Ersun Yanal to replace Vahid Halilhodzic.
Trabzonspor's latest triumph was a 4-1 thumping of Genclerbirligi on Monday, which saw Oscar Cardozo net a hat-trick.
Balikesirspor remain rooted to the foot of the table following a 4-0 hammering at relegation rivals Kayseri Erciyesspor, while Mersin Idmanyurdu's impressive season continued with a 2-0 victory over Sivasspor to stay fourth.
Elsewhere, Akhisar Belediyespor's clash with Konyaspor and Bursaspor's trip to Istanbul Basaksehir ended in goalless draws, while Kasimpasa picked up a 3-1 win against Rizespor.
