Manuel Lanzini put River ahead after only four minutes, but Carlos Luna levelled in the 29th minute. Hosts Rosario had Paulo Ferrari sent off in the 82nd minute, but they managed to hold on. Both sides are now on four points after two matches, with Estudiantes the only side to have won both their opening games.
Diego Vera scored twice in the first half as Atletico Rafaela joined Rosario and River on four points with a 3-0 win at Lanus. Cristian Canuhe added the third goal 20 minutes from time.
Belgrano Cordoba also have one win and a draw after coming from behind to win 3-2 at Boca Juniors. Juan Forlin put the hosts in front after seven minutes, but Belgrano turned the match around in the second half.
Emiliano Rigoni equalised two minutes after the break, Luciano Lollo made it 2-1 with a 65th-minute penalty and Cesar Pereyra added a third in the 90th minute. Joel Acosta struck even later to make it 3-2, but it was too little too late for Boca.
Gimnasia La Plata were 3-0 winners at home to Newell's Old Boys, with Ricardo Diaz (nine), Juan Blengio (60) and Facundo Pereyra (81) on target. Elsewhere, Colon Santa Fe beat Argentinos Juniors 1-0 thanks to Gabriel Graciani's 55th-minute goal.