The Briton, seeded third for the upcoming grass-court grand slam, bounced back from his surprise second-round exit to Radek Stepanek at the Aegon Championships earlier this month with a 6-2 7-6 (7-1) success at the exhibition event.
Murray, who played at the Hurlingham Club as part of his preparations for last year's Wimbledon triumph, was too strong for the world number 22 in a one-sided first set.
But Robredo found his rhythm in the second and the Spaniard forced a tie-break before eventually succumbing against the home favourite.
Grigor Dimitrov, who won the Queen's Club tournament after Murray's early loss, tasted defeat in London as he was beaten 7-5 6-4 by Tomas Berdych.
Bulgarian Dimitrov has never gone past the second round at Wimbledon, while Berdych was a beaten finalist there in 2010.