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.
Thiago ghosted behind defender Alejandro Lembo to head home the second from an accurate cross from the right by midfielder Fabricio.
He completed his hat-trick when he raced onto a fine ball from midfielder Henrique and beat Munoz.
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.
Universidad de Chile snatched a 1-0 win away to Alianza Lima at the Monumental in the Peruvian capital with a 86th-minute goal by substitute forward Diego Rivarola in the second match of the night.
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.
Cruzeiro were the second of the five Brazilian teams in the last 16 to win this week after Flamengo beat fellow Brazilians Corinthians 1-0 with an Adriano penalty at the Maracana on Wednesday.
Sao Paulo were held 0-0 by Peruvian champions Universitario in Lima, while Internacional lost 3-1 to Argentina's Banfield in Buenos Aires.
The first knockout round kicked off on Tuesday when Argentine sides Estudiantes and Velez Sarsfield travelled to Mexico. Holders Estudiantes beat San Luis 1-0 but Velez were hammered 3-0 by Guadalajara.
