The pair both wasted chances to seal the set as they mixed glorious winners with nervy misses, with Robredo blowing a fifth match point at 7-6, before Murray eventually blasted an unstoppable forehand to take it 11-9 and force a decider.
World number 11 Murray then broke Robredo three times in the third with the Spaniard, also chasing a first win of the year, appearing hampered by injury as he failed to chase down balls.
"Today was obviously an incredibly tough match...and I got lucky at the end of the second set," Murray said in a courtside interview.
"I fought hard and I tried my best and thankfully managed to turn it around.
"Tommy had a great tournament, he probably deserved to win the match today. He had the opportunities in the second set but sometimes that happens in sport. I just tried to fight to the end."
The victory at the $590,230 outdoor hardcourt event boosted Murray's hopes of making a sixth appearance at the season-ending ATP World Tour Finals in London in November after he was forced to withdraw from the event last year through injury.
The 27-year-old Scot moved up to 10th by adding 250 points to his tally and is within 105 points of Czech Tomas Berdych in the eight and final qualifying spot.
The victory, the 29th of his career, came after a difficult week for the normally shy Briton, who received horrendous abuse on social media for supporting Scottish independence in a referendum earlier this month.
"I want to thank my friends and family back home especially my girlfriend, she supported me a lot this week, it's been a very tough week for me and they have supported me," said an emotional Murray.
A despondent Robredo, who had bullied the Briton with a bruising forehand for the opening two sets, took little satisfaction from his second runners-up finish of the year.
"It's tough to accept when you are so close," Robredo said during the prize giving.
"Anyway Congratulations Andy, he did a great job he fought to the end and at the end his deserved it."
(Writing by Patrick Johnston in Singapore; Editing by John O'Brien/Amlan Chakraborty)
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