River's impressive win comes just three weeks before they host arch-rivals Boca Juniors at the Monumental in the first Superclasico since River lost 2-0 at La Bombonera in May 2011 and were relegated a month later.
River's last home win in the top flight was a 1-0 victory over Banfield in April 2011. They lost their next home game to Sunday's opponents Godoy Cruz and began the downhill slide towards relegation.
River, who spent last season in the second tier, had gone 10 top-flight matches without a win at the Monumental - six at the end of the 2010/11 season and the first four of this season.
But they have suddenly clicked, scoring nine goals in two matches while conceding none, adding Sunday's five-goal spree to last weekend's 4-0 victory at title holders Arsenal.
With each successive goal on Sunday, a heaving Monumental chanted River's favourite anti-Boca songs anticipating their next home match - the biggest club game on the Argentine football calendar.
Boca's 3-1 loss at Belgrano in Cordoba earlier in the day made Sunday's victory all the sweeter for River's fans.
Despite the margin of victory, River rode their luck at times with Godoy Cruz hitting the bar in the third minute and coming dangerously close with another early chance.
Coach Matias Almeyda appears to have finally hit on a successful counter-attacking formula which brought an opening goal in the ninth minute for midfielder Martin Aguirre from Rodrigo Mora's low cross to the near post.
River's key player, central midfielder Leonardo Ponzio, then marked his 100th game for the club with a cheeky goal from a free-kick 35 yards out three minutes later.
With goalkeeper Nelson Ibanez expecting a cross into the middle Ponzio found the top corner of the net with a curling shot to put River 2-0 up.
"We deserved something like this at our ground. We won well against tough opponents," Almeyda said.
River lost Ponzio with a twisted ankle on the stroke of half-time but retained their shape with Aguirre moving into the centre and young playmaker Manuel Lanzini coming in on the left.
Winger Carlos Sanchez ran onto substitute Rogelio Funes Mori's pass to score the third just past the hour and Mora netted the fourth in the 63rd minute after Ibanez spilled a cross.
Sanchez saved the best for last, lobbing Ibanez from the edge of the box in the 75th minute as River climbed to eighth with 15 points from 10 games just past the halfway mark in the 19-match Inicial, the first of two tournaments in the season.
Newell's Old Boys are top with 22 points after their 1-0 home win over Velez Sarsfield thanks to Maxi Rodriguez's goal after 30 seconds. Racing Club are second with 20 points, Belgrano third with 19 and Velez and Boca fourth with 17.