Boca's first win in four matches allowed arch-rivals River Plate to take the outright lead in the Argentine Apertura championship despite being held 0-0 by title holders Argentinos Juniors at their Diego Armando Maradona ground.
River have 10 points from four matches, one point more than Velez and Arsenal, with San Lorenzo, last season's Apertura winners Banfield and Godoy Cruz on eight.
Borghi discovered what the pressure could be after bad results at Argentina's most popular club following Boca's shock 2-0 defeat by promoted All Boys last weekend and said Sunday's match with Velez would be decisive.
Veteran midfield battler Sebastian Battaglia gave Boca the lead eight minutes before half-time having had a drive from outside the box tipped over by goalkeeper German Montoya.
From the corner, striker Lucas Viatri had his header tipped on to the bar and Battaglia was on hand to tap the ball in from less than a metre.
There was renewed fight in Boca after some tepid play in two successive defeats and Viatri took his championship tally to three when he headed Boca's second midway through the second half.
DRAMATIC FINISH
In a dramatic finish, Uruguayan striker Santiago Silva headed Velez back into the match and Boca goalkeeper Cristian Lucchetti prevented an own-goal equaliser by Viatri who was defending at a stoppage-time corner.
Independiente, another of Argentina's traditional Big Five, thought they had ended a poor start to the championship with a 1-0 win in midweek over Argentinos in the Copa Sudamericana, the Europa League equivalent in South America.
Instead, Daniel Garnero's team are bottom with one point after they were ambushed 2-1 at home by modest Arsenal, prompting fans to demand the return of last season's coach Americo Gallego who did not have his contract renewed.
Arsenal midfielder Gonzalo Choy struck a superb second goal from the tightest of angles at the edge of the box less than half a metre from the goal line.
Lanus coach Luis Zubeldia, whose side were held 0-0 by Banfield in their suburban derby on Saturday, became the first victim of the new rule where teams must return to the field for the second half within 15 minutes and was sent to the stands.
