The Napoli man came off the bench to rescue a point in the second half, after an awful mistake from the center back had raised the visitors' hopes of recording a shock victory.
The Verde striker had been let in by a slack piece of defending by Martin Demichelis, which threatened momentarily to condemn the Albiceleste to a second straight defeat.
Playing on a glorious spring day in River Plate's famous Monumental home, the first 20 minutes were forgettable for the hosts as the spirited Bolivians did an excellent job of shutting the game down. The away side played with 10 men behind the ball and only Martins up front, and despite dominating possession chances were few and far between in an uninspiring start.
With 22 minutes played, however, it seemed like the encounter had finally opened up.
A storming run from Lionel Messi left the Verde defense in his wake and Gonzalo Higuain then fired into the net to finish off a perfectly weighted through ball. Joy soon turned to indignation though. Referee Carlos Vera had signaled a free kick for the Albiceleste on the edge of the area due to late tackle on Messi, inexplicably declining to play a clear advantage.
That play was the beginning of a spell of Argentine dominance, although it could not translate this into goals. Messi took advantage of good work by Fernando Gago and Javier Pastore to release a low shot, which goalkeeper Carlos Arias was equal to.
The Paris Saint-Germain man then thought he had opened the scoring with a magnificent solo strike. Pastore did brilliantly to open up play on the edge of the area before releasing a swerving effort, but Arias' fingertips pushed the ball onto the inside of the post to ensure Bolivia would go into half time outplayed, but still level.
Ten minutes into the second half and with the Argentines striving to open the game. They were hit with a hammer blow. Demichelis was caught dithering in defense and lost the ball and Martins was in no mood to pardon the Malaga man as he blasted home.
Seconds later the Albiceleste thought they had the equalizer, but Ronald Rivero cleared off the line to deny Pastore.
The leveler, however, did not take long to arrive. Lavezzi had been on the field barely a minute when he picked up Gago's pass on the diagonal and drove home with a perfect strike past Arias from the edge of the area.
The Argentines pushed more and more men forward to try and take the initiative, but after an initial whirlwind of activity the excellent Bolivian backline stabilised, continuing their incessant pressuring spearheaded by Jaime Robles and Christian Vargas.
Messi, nevertheless, had a clear chance on the edge of the area, but uncharacteristically fluffed his lines to send the ball well wide.
There would be no winner despite strong pressure from the hosts and Bolivia earned their first point of the World Cup qualifying competition. Argentina, meanwhile, has four points from its opening three fixtures and will play next Tuesday away to Colombia.