Argentine-born Matias Vuoso scored twice for Mexico while Bolivia defender Juan Manuel Pena, making his international comeback after an 18-month absence, was sent off after half an hour for a tackle from behind.
The win was a tonic for former England coach Eriksson after a series of bad results in 2010 World Cup qualifiers.
Beaten by Honduras and Jamaica in the semi-stage of the CONCACAF qualifiers last year, Mexico began the final round in February by losing 2-0 to arch-rivals the United States.
They host Costa Rica and visit Honduras later this month in matches which are seen as key to Eriksson's future.
Mexico were missing their European-based players as the match was not played on an official FIFA date for internationals. Eriksson used the chance to test some alternatives.
Vuoso opened the scoring with a left-foot shot after 24 minutes following a cross by Israel Martinez and Brazilian-born Leandro Augusto added the second from a free kick eight minutes later.
Martinez set up Vuoso again in the 59th minute before Didi Torrico pulled one back for Bolivia, who are ninth in the 10-team South American group, in the 69th minute.
Sergio Santana won and converted a 71st minute penalty to restore Mexico's three-goal lead before debutant Jose Maria Cardenas completed the rout with six minutes left.
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