Pulu, who joined the Canberra-based side this season with an eye on forcing his way into Michael Cheika's Wallabies squad, lasted less than 10 minutes with his new club against the Melbourne Rebels before he suffered the injury on Friday.
The 30-year-old, who was born in the United States and played rugby sevens for Niue, can represent Australia through his parents.
Once dubbed the fastest man in New Zealand rugby, Pulu was invited to a pre-season camp by Cheika as he assessed players he was interested in ahead of this year's showpiece in Japan.
Brumbies team mate David Pocock also suffered an early exit from Friday's game after he suffered another head injury.
The Wallabies flanker, a key part of Cheika's World Cup plans, was inadvertently struck in the head by Dane Haylett-Petty's elbow while making a tackle and was taken off for precautionary checks.
The loose forward failed a Head Injury Assessment (HIA) and did not return to the field in the Rebels' 34-27 Super Rugby victory at Canberra Stadium.
Brumbies coach Dan McKellar told reporters they would need to carefully manage Pocock's return to play given he also suffered head and neck issues last year and was ruled out of the Wallabies final 2018 test against England.
"He failed his HIA, so there is a concussion there," McKellar told reporters. "He's obviously had a concussion on the spring tour as well so we'll just do what's in the best interest of Dave."
Pocock may have to sit out the Brumbies' next Super Rugby game against the Chiefs, while veteran winger Henry Speight is also a doubt after he suffered a hip injury.