Chelsea’s Enzo Fernandez opened the scoring, with further strikes from Nicolas Tagliafico and Nicolas Gonzalez sealing victory in La Paz.
Brazil also secured their second win after beating Peru 1-0 thanks to a 90th-minute goal from Marquinhos.
Marcelo Bielsa suffered his first defeat in charge of Uruguay after Ecuador came from behind to win 2-1.
Bielsa’s side, who beat Chile on Saturday, took the lead through Agustin Canobbio, but a goal in each half from Felix Torres secured the win for Ecuador, who also missed a penalty through Enner Valencia.
Messi watched Argentina’s victory from the bench but was not listed among the substitutes having complained of tiredness after scoring the only goal in Wednesday’s win against Ecuador.
“Leo was not ready to play,” said boss Lionel Scaloni after the game. “He tried to recover, but he did not feel comfortable.”
Former Manchester United midfielder Angel di Maria captained the side in Messi’s absence and provided the cross for Fernandez to stroke home a first-half opener.
Bolivia forward Roberto Fernandez was then sent off for a high challenge on Cristian Romero, before Tagliafico headed home Di Maria’s free-kick before the break.
Julian Alvarez hit the post in the second half as the world champions continued to dominate, before Gonzalez wrapped up the win with a left-footed drive late on.
Brazil are the only other side in South American qualifying to make a perfect start but were frustrated for large periods by a defiant Peru side in Lima.
The game looked to be heading for a draw before Neymar’s in-swinging corner was glanced home at the near post by by Marquinhos.
Elsewhere, Chile and Colombia played out a goalless draw in Santiago, while a stoppage-time penalty from Salomon Rondon secured a 1-0 victory for Venezuela against Paraguay.
