Barba scored his second try of the night in almost identical fashion to his first when a James Graham pass from a scrum on the Bulldogs 20m line sent Josh Jackson upfield and he dished off to a flying Barba who raced 45m for a converted try and for the Dogs to regain the lead 12-6 after 29 minutes.
Six minutes later Inu outjumped Linnett chasing a Reynolds kick for a benefit-of-the-doubt try and, after the former Warrior converted, the Bulldogs had stretched their lead to 12.
The Cowboys opened the scoring in the second half eight minutes in when Antonio Winterstein chased through a Thurston kick that bounced away from Barba to close the gap to 18-12 with Thurston's conversion.
Matt Bowen produced a try saver on Brett Morris in the 53rd minute before Thurston dummied his way over from close range in the 62nd minute for his first try of 2012.
Thurston converted to tie the scores up at 18-18, but the Bulldogs didn't panic and duly sealed victory.
-AAP