The visitors salvaged a point in the dying seconds in Montevideo to maintain their impressive start to the Clausura season.
River had seemed on course for victory after 32-year-old forward Sebastian Taborda gave them the lead in the 29th minute.
But Fenix mustered a late response as Maximiliano Perez scored an equaliser in second-half stoppage time.
Fenix share top spot with Penarol and Danubio, with the trio one point clear of the Montevideo Wanderers, who edged giants Nacional 3-2 in a thrilling encounter.
Gaston Periero's early goal was cancelled out by Rodrigo Pastorini for the Wanderers, who then moved in front through Mauricio Gomez.
Juan Cruz Mascia drew Nacional level four minutes before half-time, but Matias Santos' second-half goal earned victory for the Wanderers.
Reigning Clausura champions Defensor Sporting conceded a lead in a disappointing 3-1 home loss to Cerro.
After a goalless first half, Defensor went ahead in the 46th minute thanks to Ramon Arias' strike.
However, Defensor then collapsed in search of a fourth successive win in all competitions.
Carlos Grossmuller and Hector Acuna each scored penalties to put Cerro ahead, before the latter added a third goal with 10 minutes remaining.
Octavio Rivero scored a hat-trick as Rentistas beat Cerro Largo 3-1 on Sunday, while Jonathan Rodriguez's second-half double gave 10-man Penarol a 4-3 win at El Tanque Sisley.
Elsewhere, Danubio beat Miramar Misiones 3-1, Sud America edged Liverpool 2-1, and Racing and Juventud played out a goalless draw.
