Arsenal took the lead in the 10th minute as defender Clemente Rodriguez converted a Nicolas Aguirre cross from the left into his own net.
Pablo Mouche equalised 18 minutes later with a right-foot strike before Pablo Ledesma gave Boca the win in the 68th minute.
It gave six-time Libertadores winners Boca their first win from three group games.
Mouche told Fox Sports: "This is a key victory for us. It was a must-win game for us.
"We had the necessary attitude to overcome going a goal down."
Group leaders Fluminense of Brazil beat Venezuela's Zamora 1-0 to make it three wins out of three.
Anderson netted the match's only goal in the 57th minute with a shot from outside the area.
The defender told Globoesporte.com: "It is too hard when all 11 players of the other team play behind the ball.
"I'm not a hero at all. Maybe the team might be called a hero. A 1-0 score is great in this kind of game."
In Asuncion, hosts Libertad played out a 1-1 draw with Brazil's Vasco da Gama, who took the lead thanks to a 16th-minute header from Diego Souza.
Souza elbowed an opponent and was sent off seven minutes after the interval, allowing Libertad to level in the 70th minute through Jose Nunez.
Libertad also ended the game with 10 men as Nunez tackled Rodolfo and was shown a second yellow card.
Vasco defender Dede told Globo TV: "We lost two points. This is not what we were looking for. It was a good chance for us."
Finally in Mexico, Cruz Azul and Corinthians played out a goalless draw.
Cruz Azul lead Group Six on seven points, two ahead of Corinthians.