One of nine changes from the weekend win over Everton, Welbeck was on target twice before a splendid Alan Judge free-kick set up a tense final half hour at Emirates Stadium.
The Championship side were emboldened and, for a period, pressed hard in search of an equaliser that ultimately never came, Arsenal stabilising adequately and eventually sealing Unai Emery's sixth straight victory through substitute Alexandre Lacazette in stoppage time.
The new Gunners boss is attempting to do something his predecessor Arsene Wenger never achieved in winning this competition, and the quality of several passages in between the stern second-half examination will have delivered belief that is very much a possibility.
The course of the match might have differed had Bernd Leno's puzzlingly poor touch from a Shkodran Mustafi back-pass not trickled wide of his own net in the second minute.
Instead, Arsenal were ahead moments later as Welbeck rose unchallenged to emphatically head in Matteo Guendouzi's corner in the fifth minute.
His second, eight minutes before the interval, capped a sweeping move with Henrikh Mkhitaryan, Alex Iwobi and the overlapping Nacho Monreal combining to provide a tap-in.
An unexpected twist came when midfielder Judge fizzed a brilliant free-kick in off the upright and Brentford almost had an equaliser in quick time, Sergi Canos flashing a shot over the bar.
Penalties remained a possibility right until Lacazette put the seal on Arsenal's victory with a determined finish three minutes into added time, fellow substitute Lucas Torreira with the assist.
What it means: Gunners in sync
This kind of banana skin fixture could well have eroded some of Arsenal's momentum, but Emery's second string handled the occasion professionally when required and, at times, with real style.
Emery's decision to retain only Monreal and Mustafi on his starting XI was justified as the fluidity of the second goal demonstrated an entire squad operating on the same wavelength.
Leno looks the part
Though beaten by Judge's free-kick - for which he could do very little - two moments in the first half served to illustrate how comfortably Leno can slot in should Petr Cech's form tail off.
The German recovered from a nervy early moment to instigate Welbeck's second with a brave pass from the back, having not long earlier raced off his line to snuff out a counter-attack in true sweeper-keeper fashion.
Guendouzi lapse punished
Supplier of the opening goal, Guendouzi undid that good work by offering Brentford a way back into the match after half-time. It was the teenager's unnecessary foul on the edge of the area which Judge emphatically converted.