The 23rd-minute goal and Boca's victory celebrations met with an eerie silence from the crowd of 60,000 at the Monumental stadium, with away fans banned at first division matches since the end of last season in an effort to curb hooligan violence.
Boca, who have gone four matches without conceding a goal and picked up 10 points in the process, are on 19, four behind leaders and title-holders Newell's Old Boys halfway through the first of two championships in the season.
Newell's won 2-0 at Quilmes on Saturday with San Lorenzo second on 20 points after a 0-0 draw at home to Tigre.
River dropped to seventh, nine points off the pace, after Racing Club's woeful season worsened with a 1-0 home loss at El Cilindro to Atletico Rafaela, who are now fifth. Racing are bottom with two points and have a 3-18 goals record in 10 matches.
Boca, who weathered an early storm of River attacks, went ahead when Juan Roman Riquelme sent a pass out wide to the right, Juan Manuel Martinez crossed low and Gigliotti deflected the ball past goalkeeper Marcelo Barovero into the net.
"We didn't want Newell's to get away from us. We're going to fight for the title to the end," former Argentina playmaker Riquelme told reporters.
River coach Ramon Diaz said: "Today the best team didn't win ... A fairer result would have been a draw.
"The Argentine championship always gives you a chance for revenge. Next Sunday we're going to visit Newell's in Rosario ... and if we can't bring Newell's down, then who will?"
Both sides hit the post, Martinez towards the end of the first half and River substitute Rodrigo Mora with a header in the dying minutes as the visitors defended their lead.
River had more possession and created some good moves but were guilty of errors with the final ball against a generally tight defence.