Defender Bilel Mohsni also scored on his debut with fellow centre-half McCulloch's hat-trick turning the game in the home side's favour before Andy Little put the cherry on top of the Light Blues' cake.
Minnows Arbroath had held Celtic in Parkhead in last season's Scottish Cup on their last visit to Glasgow, and held hopes of at least restricting a very experienced-looking Rangers side chasing a fifth win from five Scottish League One games.
McCulloch almost made an impact after only six minutes following Arnold Peralta's corner but the Gers captain was denied by Arbroath goalkeeper Scott Morrison after a scramble.
Having withstood that early scare the Gayfield side grew in confidence and in
the 13th minute only Mohsni's intervention to turn the ball past his own post
prevented Cook's low cross reaching on-loan Motherwell man Lee Erwin.
Arbroath passed up the chance of the half after 20 minutes and Cook was again the provider from the left but his drilled ball across the six-yard box was just too far in front of Erwin's lunge as Bell's goal begged a touch.
Rangers' only threats came from set-pieces and from another Peralta corner Mohsni powered a header over the bar.
The home fans' expectations had turned to frustration by the 35th minute and their feelings were vented on referee Crawford Allan, who waved aside hopeful appeals for a penalty as Jon Daly's run to meet a cross were impeded by Hamilton.
And the visitors silenced the Glasgow crowd two minutes after the restart when Peralta gave away possession on the touchline for Ross Chisholm to feed the ball to Erwin he laid it off to Cook whose measured chip carried the ball over keeper Cameron Bell and under his crossbar to stun the home crowd.
Rangers came within inches of an equaliser when an inswinging free-kick from Steven Smith was almost turned into his own net by Michael Travis under pressure from Daly.
However, the equaliser was only delayed until the 57th minute from another
Smith deadball, which was cleared only as far as Nicky Law.
Law's shot from just outside box took a deflection to the feet of Mohsni, who calmly side-footed past the helpless keeper for a dream debut goal.
The Gers then played with a bit more urgency and David Templeton, on for
Nicky Clark, dribbled past four opponents before firing his drive just over the bar.
McCulloch powered his side into the lead after 65 minutes from Law's corner before Peralta's free-kick rebounded off the Arbroath wall 10 minutes later and, from 30 yards, the skipper drilled a superb shot high into the top corner.
That wrapped up the points but not the scoring and two minutes from time Templeton sent Northern Ireland forward Little through and his angled drive past Morrison made it 4-1.
Then, in injury time, veteran McCulloch finished the rout to claim his second straight treble by nodding home Peralta's cross at the back post.