Jevons scored a superb first-half goal to add to Marvin Elliott's soft opener in the first round clash and then made it three with another flashing finish four minutes from time.
It was a convincing victory for the Championship side, who never looked in danger against their League Two rivals even though the home side had plenty of possession.
Alan Connell had Brentford's best two chances but failed to score with either, while Sammy Moore's long-range effort was well saved by City keeper Chris Weale in the first half after the Bees started brightly.
Butcher's new-look team played some fluid football at stages and created plenty of efforts on goal, but they couldn't find the net and the game was all but over when they went into the break already 2-0 down.
Connell wasted an excellent chance early on when he shot wide of an open goal and it proved a costly miss because City took the lead after 16 minutes.
Poor marking allowed a corner to reach Elliott and his soft header bobbled off defender Grant Basey and into the net.
If that goal was soft, Bristol's second was superb as Jevons lashed the ball high into the net after 42 minutes following good work from rampaging defender Liam Fontaine.
Brentford had a lot of possession in the second half and saw Connell's header saved on the line.
But when midfielder Paul Brooker had a row with boss Butcher and was immediately subbed, the writing was on the wall for the Bees.
City proved their superiority in the dying seconds when another free-kick reached Jevons unmarked and he thumped the ball home on 86 minutes.