The Robins were in control of this one from start to finish and could have won by a far wider margin but a heroic performance by Jake Hayhoe, the Bury Town goalkeeper, prevented a defeat well into in double figures.
Harry Williams enhanced his growing reputation with yet another hat-trick, taking his tally to 29 for the season while Zack Kotwica added three of his own.
Left back James Bowen claimed a brace and Bobbie Dale completed the scoring
The Suffolk side, who ply their trade in the Ryman League Premier Division, made the 350-mile round after they were reinstated to the competition following their 3-0 defeat by Dagenham and Redbridge in round one.
The east Londoners were found in breach of FA Youth Cup competition rules for playing an ineligible player, West Ham United loanee Thomas Gogo, and were removed from the competition.
Russell Milton's impressive young side hit VCD Athletic for seven in round one, found five against Newport County at the weekend and took their tally to 21 goals in three games here.
Their reward is a third round tie against Birmingham City in round three.
Cheltenham started positively against their non-league opponents and were in front as early as the fourth minute.
Williams grabbed the opener, meeting Adam Powell's cross with a perfect header to register the first goal of the night.
Powell forced a diving save from 20 yards a moment later, while Williams wasted a chance to double the advantage with 12 minutes played.
Cheltenham were comfortable after the goal without really stretching the Suffolk side, and were almost made to pay for some sloppy play on 29 minutes when Noel Aitkens' long ranger brought about a notable save by Harvey Rivers.
But Kowica scored the second a couple of minutes later, racing onto Rivers' long punt upfield before sliding an effort across Jake Hayhoe, the Bury Town number one.
Bowen added the third four minutes later after a goalmouth scramble sparked by Powell's free-kick from the left. Williams' initial header was saved but the young left back was on hand on bundle it home at the second attempt.
Two goals in quick succession ended the tie as a contest shortly after the restart.
Dale made it four after 49 minutes, running into the path Powell's angled ball before clipping the ball across Hayhoe.
Then two of the three players that have featured for Mark Yates' first team this term combined for the fifth only three minutes later.
Kotwica burst to the byline and cut one back towards Williams who accepted the simple chance to register his second of the evening.
Number six arrived on 63 minutes, Powell again the supplier from the right and Bowen with the header from 10 yards.
Williams completed his hat-trick with 20 minutes remaining with a close-range header after Powell delivered a low corner kick from the right.
Kotwica notched the eighth after accelerating into the box before smashing it home from the angle, and completed his hat-trick from the penalty spot with three minutes left.
CHELTENHAM TOWN: Rivers (H Reynolds 62); Sheppard, Hamilton ©, Keightley, Bowen; Kotwica, Whitehead, Powell, Kirkpatrick-Jones (D Reynolds 70); Dale, Williams (Goodwin 74). Subs not used: Daly, Craddock.
BURY TOWN: Hayhoe; Boulter (Wigley 79), Potter ©, White, Coldron; Call (Bedingham 63), Allen, Aitkens, Thorogood; Jollans. Subs not used: Seymour, Wigley.
REFEREE: M Rushton