The British number one looked to be in considerable trouble when Andujar followed up the first set with a break in the first game of the second.
However, Murray rallied and rattled off the remaining six games of the set to comfortably send the match to a decider.
Earlier, top seed David Ferrer opened his campaign with a regulation 6-2 6-3 win over Mikhail Kukushkin of Kazakhstan.
Ferrer, a three-times champion on the clay of Acapulco between 2010 and 2012 and runner-up last year, took an hour and 27 minutes to seal victory, although he converted just three of 16 break points.
The Spaniard will next play his compatriot Feliciano Lopez, who saw off Frenchman Edouard Roger-Vasselin 6-4 6-2.
"I'm not thinking about the final, I'm only in the second round", Ferrer said. "Now I'm only thinking about playing my best tennis against Feliciano Lopez, he's a good player I need to give my best to beat him".
Ferrer has been in fine form since the start of the Latin American claycourt swing, winning the tournament in Buenos Aires and reaching the semi-finals in Rio de Janeiro last week.
"Playing in Buenos Aires was not in my plans", said Ferrer who replaced Rafael Nadal in Argentina. "I'm tired, but I will not be making any excuses, I need to take a rest now and get well for the rest of the tournament".
John Isner fell at the first hurdle as the third seed dropped to a 7-6(4) 7-6(5) defeat to Croatia's Ivo Karlovic in a battle of the big-servers that saw Croatian Karlovic fire down 15 aces to the American's eight.
The match was settled by two tie-breaks, meaning the pair are now tied for the record of most consecutive tie-break sets played, matching Ivan Ljubicic and Marat Safin's eight in a row.
There were also wins for high seeds Grigor Dimitrov, Kevin Anderson, Gilles Simon and Ernests Gulbis, but eighth seed Vasek Pospisil was bounced out by Alexandr Dolgopolov 6-2 6-2.
Acapulco International results
2-Andy Murray (Britain) beat Pablo Andujar (Spain) 3-6 6-1 6-2
Ivo Karlovic (Croatia) beat 3-John Isner (U.S.) 7-6(4) 7-6(5)
4-Grigor Dimitrov (Bulgaria) beat Marinko Matosevic (Australia) 7-5 6-1
5-Kevin Anderson (South Africa) beat Stephane Robert (France) 6-2 7-6(7)
6-Gilles Simon (France) beat Tim Smyczek (U.S.) 6-1 6-2
7-Ernests Gulbis (Latvia) beat Lu Yen-Hsun (Taiwan) 6-1 6-1
Alexandr Dolgopolov (Ukraine) beat 8-Vasek Pospisil (Canada) 6-2 6-2
Feliciano Lopez (Spain) beat Edouard Roger-Vasselin (France) 6-4 6-2
David Goffin (Belgium) beat Lukasz Kubot (Poland) 6-4 2-6 6-4
Marcos Baghdatis (Cyprus) beat Miguel Angel Reyes (Mexico) 6-2 6-4
Donald Young (U.S.) beat Jarkko Nieminen (Finland) 6-3 6-3