An action-packed match saw two red cards, 10 yellow cards and three goals of high quality, but ultimately it was Boca who came out on top to make it eight wins from eight in the league and open up a nine-point lead at the top of the table.
The result caps off a devastating week for River, who were beaten 4-0 in the league last weekend, dumped out of the Copa Libertadores in the semi-finals in midweek and have now lost at home to their most hated enemies to fall 12 points adrift of the pace-setters.
Rossi almost gifted the hosts the opening goal after 20 minutes when his clearance only found Gonzalo Martinez, but the River Plate midfielder could not make the most of it and blazed his subsequent effort well over the crossbar.
Martinez threatened again shortly before the half-hour mark when he collected the ball inside the box before beating one defender, only to see his eventual shot blocked and go behind for a corner.
River Plate continued to look the most likely to break the deadlock, and Rossi again almost gave them a helping hand when his quick throw was picked off by Ariel Rojas, although the Boca keeper atoned for his error by saving the following strike.
The hosts' momentum was stopped in its tracks with just seven minutes of the first half remaining, though, as Ignacio Fernandez was shown a straight red card for a high boot on Cardona, catching the Boca midfielder in the chest.
Things went from bad to worse from the resulting free kick too as Cardona dusted himself off and fired a stunning strike past Lux to give Boca the lead in the closing stages of the first half.
Boca almost doubled their advantage before the break too when Cristian Pavon's dipping strike was pushed away, but it was River who came out the strongest in the second half despite their numerical disadvantage.
Scocco and half-time sub Nicolas de la Cruz both squandered chances to level things up before an equaliser of sorts did occur when Cardona was given his marching orders to make it 10 against 10.
Referee Nestor Pitana had little choice with the first sending-off, but the second was incredibly harsh as Cardona was adjudged to have elbowed his marker in the face when there was minimal contact.
As was the case with Boca, River Plate did not need long to capitalise on the red card and they restored parity in style with just over 20 minutes remaining when Leonardo Ponzio arrowed a spectacular 30-yard effort past a helpless Rossi.
It was a deserved equaliser and at that stage it appeared as though River would be favourites to go on and win the game, but they were level for only four minutes before Boca regained the lead against the run of play.
Nahitan Nandez found space on the right side of the area to latch on to a clipped pass over the top, volleying his first-time finish past the keeper to silence El Monumental once again.
That proved to be the winning goal as the champions piled more misery on River Plate, who have now lost three consecutive matches across all competitions for the first time since January.
Boca, by contrast, are in their best form in more than a decade having racked up a perfect record from their opening eight fixtures, setting them well on course for a 33rd Argentine league title.