The league leaders had scored just 12 goals in their first nine league matches, with their defence - only four goals conceded - seeing them remain undefeated.
But Rivero's sending off after just five minutes allowed Guillermo Sanguinetti's men to open up the visitors to the Estadio Alejandro Villanueva.
Christian Cueva broke through in the 18th minute, shooting on the turn on the edge of the penalty area and beating Cienciano goalkeeper Diego Morales at his near post.
Pablo Miguez doubled Alianza's lead in the 32nd minute, nodding home from a cross, before he bagged a brace 10 minutes later - lashing home a left-foot volley after Miguel Araujo pulled a ball back into the danger area from the right byline.
Walter Ibanez made it 4-0 just prior to half-time, converting a spot kick straight down the middle of the Cienciano goal, while Mauricio Montes was on hand to tap home Alianza's fifth on the hour mark, sliding in at the back post after Victor Cedron missed Gabriel Costa's cross.
Sporting Cristal remained within four points of the leaders, and hold a game in hand, after a 3-1 win at Leon de Huanuco.
Sergio Blanco's first-half brace helped set up a fourth straight victory for Daniel Ahmed's men.
Level with Cristal on 20 points but third on goal difference, Melgar enjoyed a 2-0 win on the road at Universidad San Martin.
Union Comercio dropped off the pace set by the top three, falling four points behind Melgar after a 2-1 loss away to mid-table Sport Huancayo.
Real Garcilaso climbed to fifth, two points behind Comercio, after a 2-0 win over Universitario - who they joined on 14 points.
Juan Aurich are also on 14 points but seventh on goal difference, after striking twice early in a 2-0 win over UTC Cajamarca.
Janio Posito's quick-fire brace late on helped Los Caimanes to a 3-2 win away at Cesar Vallejo, catapulting the victors to 10th.
And just a week after opening their winning account, San Simon made it two from two - scoring inside four minutes on their way to a 1-0 victory over Inti Gas to move level on points with second-bottom Universidad San Martin.
