Defending champions Boca Juniors also had two players dismissed in a 2-0 defeat at lowly Independiente.
Luciano Leguizamon put Arsenal ahead from a penalty in first-half stoppage time after a foul by Nicolas Sanchez, who was sent off.
Despite being a man down, River replied with a Radamel Falcao Garcia header and a delightful chip from Gallardo, his first goal after returning for a third stint at the club in January, early in the second half.
Arsenal had Sergio Sena sent off for a foul on Gallardo in the 73rd minute and River Plate substitute Rodrigo Archubi, who had only been on for 10 minutes, followed immediately for lashing out at an opponent.
Former AS Monaco, Paris St Germain and D.C. United forward Gallardo added a third in the 77th minute before Arsenal had defender Anibal Matellan sent off for a second bookable offence.
River climbed to fifth with 10 points from five games.
Lanus, who beat Newell's Old Boys 2-1 on Friday, lead with 12 points, one ahead of Velez Sarsfield and Colon.
Rosario Central gave new coach Reinaldo Merlo a perfect debut as they beat San Lorenzo 3-1 for their first win of the championship.
Boca were reduced to 10 men after 23 minutes against Independiente when Pablo Mouche was booked for a bad tackle and dissent in quick succession.
Independiente playmaker Daniel Montenegro scored 10 minutes after halftime, putting the ball through Fabian Vargas's legs just inside the Boca half before floating a 30-metre shot in off the post.
Montenegro added the second from a 78th minute penalty and Boca's frustration boiled over when Sebastian Battaglia was dismissed for lashing out at a player.
Independiente substitute Hernan Fredes came close to scoring an extraordinary third goal when he tried to chip Roberto Abbondanzieri from the halfway line, but his shot bounced just in front of the goal and clipped the crossbar as it went over.
