The Colombian will take on top seed Andy Murray in the quarter-finals, the Briton surviving a second-set wobble and a stiff back to beat American qualifier Tim Smyczek 6-1 4-6 6-2.
World number four Murray, playing his first tournament since his semi-final loss to Spaniard Rafa Nadal at Wimbledon earlier this month, broke the unheralded Smyczek three times in the final set to advance on a cool evening.
Left-hander Falla came from 0-3 down in the third set and needed five match points before clinching the tiebreak 10-8 against a temperamental Gulbis at the LA Tennis Center.
Gulbis, who was repeatedly warned by the umpire for racket abuse, appeared to have the match firmly within his grasp after taking early control in the third set but he then lost three successive games for Falla to level.
Although the Colombian failed to serve out when leading 5-4 and squandered a 6-4 advantage in the tiebreak, he finally wrapped up victory in two hours and 15 minutes when Gulbis netted a backhand.
"I played really well in the last set," a beaming but exhausted Falla said courtside after reaching his second ATP quarter-final of the year. "We were both tired ... and I am really happy to get through."
Asked whether he had benefited from his opponent's hot temper, Falla replied: "It helped me a lot. He gave me a few chances and I got them. I was just trying to be aggressive."
Gulbis, playing his first tournament since being sidelined in May by a hamstring injury he suffered at the French Open, felt he had paid the price for his recent inactivity.
FISHING LAYOFF
"If you don't fish for two months, maybe you're bad at fishing," the 21-year-old told reporters with a smile. "I was so tired. It was really tough for me to go for long rallies."
Gulbis was docked a point after hurling his racket to the ground in frustration after being broken to trail 3-5 in the second set and was fortunate to escape another penalty at 8-7 down in the tiebreak.
"Whatever I would do, I would get a warning," he said. "This referee, his only fun in life is to give warnings. But if he had disqualified me in the tiebreak, that would be stupid."
Murray, bidding for his first ATP title this year, cruised through the opening set in 28 minutes and appeared to be on track for a demolition job against the world number 192.
However, the Scot was unexpectedly broken three times in the second set as Smyczek levelled the match. Further service breaks were traded in the first two games of the third set before Murray restored order.
"Tonight was a very good match with a lot of long rallies, especially in the first two sets," the Scot said. "It was a good standard and hopefully it will stand me in good stead."
Asked how he had fared physically, Murray replied: "I was stiff, maybe because it was cool this evening. It's the first time I have played since Wimbledon. Hopefully I will be okay tomorrow."
Fourth-seeded Spaniard Feliciano Lopez booked his place in the last eight by beating Israeli Dudi Sela 7-6 6-4 in the first match of the day.
Lopez, who clinched his second ATP career title in Johannesburg in February, blasted 12 aces on the hardcourt surface to seal the win in one hour, 37 minutes. He will next meet American wildcard James Blake, who improved his record to 3-0 over Germany's Benjamin Becker with a 7-5 7-6 victory.
(Editing by Steve Ginsburg and Peter Rutherford)
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