Thiago scored his goals in the sixth, 21st and 41st minutes for a dominant Cruzeiro side, who aim to make up for losing last year's final against Estudiantes of Argentina.
Thiago put the Brazilian team ahead when he slipped the attentions of teenage central defender Sebastian Coates and beat Rodrigo Munoz with a low shot through the goalkeeper's legs.
Striker Mario Regueiro pulled one back for Nacional five minutes into the second half, a goal that may turn out to be crucial when the teams meet again in next week's second leg in Montevideo.
Rivarola beat both Alianza central defenders to the ball when all three went for a punt out of the Chileans' defence and raced into the box to shoot over advancing keeper George Forsyth.
Sao Paulo were held 0-0 by Peruvian champions Universitario in Lima, while Internacional lost 3-1 to Argentina's Banfield in Buenos Aires.
