Reiss Nelson's first-half effort proved to be the difference between the two sides, as Thomas Frank's men huffed and puffed for an equaliser to no avail, thanks in no small part to crucial contributions from the returning Aaron Ramsdale.
A century of appearances for Emile Smith Rowe and senior debut for Charles Sagoe Jr dominated the visitors' team news headlines, but it was a more familiar face who broke the deadlock with just eight minutes gone.
Zanka's attempted backpass to Mark Flekken was cut out by Eddie Nketiah, and the striker played the ball back across the box to Nelson, although a sliding Nathan Collins would seemingly intercept.
However, the ball squirmed under the Bees defender to Nelson, who took a touch before calmly side-footing his first shot of the night into the bottom corner from 12 yards.
Brentford's futile search for an equaliser was met with stiff resistance from Arsenal, who were close to a second with 31 minutes gone, but the foot of Flekken denied Smith Rowe's low effort after some fancy footwork from the Englishman.
A mishit Yoane Wissa attempt in the 34th minute represented Brentford's best opening of the first half, but the hosts spent the opening 20 minutes of the second period camped inside the Arsenal half, subjecting their London rivals to a wave of attacks.
Making two saves in quick succession, Ramsdale - who hitherto had little to do - kept out a Saman Ghoddos volley in the 64th minute, only one moment before tipping a fizzing Wissa strike onto the post.
Arsenal remained a threat in their own right, though, and Nelson should have done better from Kai Havertz's cutback in the 70th minute - firing wide of Flekken's goal at the back post.
Arteta's men had to survive another two heart-in-mouth moments in the 77th and 78th minutes, though, as an inspired Ramsdale parried Frank Onyeka's long-range drive away before a tight offside call denied Zanka a leveller.
Following Ramsdale's earlier heroics, Jakub Kiwior then produced a sensational block from Keane Lewis-Potter's 84th minute goal-bound effort, and Mikel Arteta's men saw out five minutes of injury time to secure a hard-fought place in the fourth-round draw.
Another away day at Bournemouth awaits Arsenal in Saturday's Premier League showdown, while the Bees seek to snap a three-game losing streak away to Nottingham Forest a day later.