Tall Uruguayan striker Sebastian Balsas headed one goal and laid on the other to sink Boca, whose 2-1 home win over title favourites Velez Sarsfield last weekend appeared to have given Borghi some breathing space.
Boca, also beaten at home by Racing Club, have managed just four points from five matches, while unbeaten San Lorenzo have 11 points, one less than new leaders Arsenal, who beat title holders Argentinos Juniors 1-0.
Juan Sebastian Veron's Estudiantes are level with River on 10 points after a 2-1 away win over Godoy Cruz on Friday night and they have a game in hand.
MEDEL OFF
Boca, who saw a first-half header by three-goal striker Lucas Viatri come back off the bar, had Chile midfielder Gary Medel sent off in the 40th minute for a second booking.
Borghi had no complaints about the red card but told reporters San Lorenzo's Colombian midfielder Leo Lopez should have been sent off earlier for a foul on Medel.
Balsas, 1.97 metres tall, rose to central defender Jonathan Botinelli's cross from the left to head the opening goal in 52nd minute with goalkeeper Cristian Lucchetti managing to divert the ball but not enough as it went in off his right hand post.
Boca central defender Matias Caruzzo might have equalised for Boca two minutes from time but goalkeeper Damian Albil brilliantly saved his header.
From the counter-attack, Balsas raced down the left and squared to Juan Menseguez who placed his shot perfectly inside the base of Lucchetti's left post.
Argentina World Cup striker Martin Palermo ended his four-match goal drought by pulling one back in the fifth minute of stoppage time when Albil spilt a cross.
River Plate also have a wary eye over their shoulders at the bottom end of the standings.
All Boys beat Olimpo 1-0 at home in a match between two promoted teams and sent River, whose relegation average over the last two seasons was among the lowest, into the drop zone.
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