Andujar, who had match points against Rafael Nadal in Rio de Janeiro a week ago, got off to another strong start as he took the first set and then broke Murray again. However, the No.2 seed responded by winning six games in a row on his way to a 3-6, 6-1, 6-2 victory.
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Murray, making his first appearance in Latin America, was due to face Joao Sousa in the second round early this morning after the Portuguese defeated France's Adrian Mannarino 1-6, 6-2, 6-2.
David Ferrer, the top seed, breezed past Mikhail Kukushkin of Kazakhstan 6-2, 6-3 to set up a second-round meeting with Feliciano Lopez. The Spaniard beat Edouard Roger- Vasselin of France 6-4,6-2.
John Isner, the No.3 seed, is out after going down 7-6 (7-4), 7-6 (7-5) to Ivo Karlovic. Fourth-seeded Grigor Dimitrov beat Australian Marinko Matoesevic 7-5, 6-1.
In the women's tournament, the top three seeds, Dominika Cibulkova, Eugenie Bouchard and Kaia Kanepi, advanced but there was an early exit for the fifth-seeded Yvonne Meusburger.
Cibulkova defeated Urszula Radwanska 6-3, 6-4 while Bouchard was a 6-2, 6-2 winner over Shahar Peer. Kanepi was taken to three sets but overcame Jana Cepelova 6-1, 3-6, 6-2. Meusburger lost in straight sets to qualifier Sharon Fichman, 6-1, 6-3.
Meanwhile, Murray will warm up for his Wimbledon title defence at the BNP Paribas Tennis Classic at London's Hurlingham Club.
Organisers have announced that, for the second successive year, Murray will be one of the star attractions at the exhibition event taking place from June 17-20. A number of veteran players, including Tim Henman, Goran Ivanisevic, Mark Philippoussis and Pat Cash, have also been confirmed.
Britain's James Ward was knocked out of the Dubai Duty Free Tennis Championships in the second round yesterday, losing 6-1, 7-6 (7-3) to Russia's Mikhail Youzhny.