Overnight leader Willett shot a final round 69 at the Emirates Golf Club to finish on 269, 19 under par, after needing to make a 15-foot putt on the last to clinch his fourth European Tour title.
"When you win like that in that fashion, it does feel that little bit extra special when the pressure is on, being able to produce the goods," said the 28-year-old, runner-up in last season's Race to Dubai.
Sullivan made up one shot on him and Cabrera-Bello to finish joint second.
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Spain's Alvaro Quiros and South Korea's An Byeong-Hun were joint fourth after carding final round 65s for a 16-under par total.
World number two Rory McIlroy, last year's winner, and Henrik Stenson finished joint sixth a further shot behind.
Willett, 28, is hoping to win a place in Europe's Ryder Cup team later this year and was neck-and-neck with Sullivan throughout the final round.
He fell a shot behind early on as his fellow Englishman birdied the second and fourth holes, but quickly recovered to lead by three.
A bogey on the 14th gave hope to his pursuers and Cabrera-Bello was only one shot behind after a birdie at the 17th.
Sullivan, playing a hole ahead, roared in delight and punched the air after sinking a long putt on the final hole for a share of the lead.
He then watched on from the scorer's area as Willett strode the final fairway before his dramatic birdie putt averted a three-man playoff and secured prize money of more than 402,000 euros ($448,591.80).
"Fair play to Danny, to birdie the last hole was a tough putt," Sullivan told reporters. "He was under a lot of pressure and stepped up well.
"I struggled again with my driving. I'm not really hitting any fairways and that made it difficult to get lots of birdies from there. I put myself in a great position to win and just not got over the line."
Spain's Alvaro Quiros and South Korea's An Byeong-Hun were joint fourth after carding final round 65s for a 16-under par total, with world number two Rory McIlroy and Henrik Stenson joint sixth a further shot behind.
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