Andujar, who held Rafael Nadal to match points in Rio de Janeiro a week ago, got off to another strong start as he took the first set and then broke Murray again, but the Scot responded by winning six games in a row on his way to a 3-6 6-1 6-2 victory.
Murray, making his first appearance in Latin American, will face Joao Sousa in the second round after the Portuguese saw off Adrian Mannarino 1-6 6-2 6-2.
Top seed David Ferrer, who won three straight titles in Acapulco from 2010 to 2012 when the tournament was played on clay, adjusted well to the new hard court as he breezed by Mikhail Kukushkin of Kazakhstan 6-2 6-3.
That has set up a second round meeting with Feliciano Lopez, with the Spaniard beating Edouard Roger-Vasselin of France 6-4 6-2.
Third seed John Isner is out already, having gone down 7-6 (7/4) 7-6 (7/5) to Ivo Karlovic, but there were no such problems for fifth seed Kevin Anderson, who beat French qualifier Stephane Robert 6-2 7-6 (9/7).
Seventh seed Ernests Gulbis dispatched Yen-Hsun Lu 6-1 6-1 while Alexandr Dolgopolov carried over his strong form from Rio by beating eighth seed Vasek Pospisil 6-2 6-2.
Fourth seed Grigor Dimitrov beat Australian Marinko Matoesevic 7-5 6-1 and Gilles Simon advanced with a 6-1 6-2 win over Tim Smyczek.