The win put them second in the table behind leaders Gravesend on points difference.
With only Eddie Armah and Jon Reid unavailable, they took the field in confident mood. And within the first five minutes, the Reds were on the back foot.
A quick take from debutant Sully left the defence flat-footed with a perfectly-timed offload putting Fairbairn into space, and the left winger surged over.
Adams slotted the extras for an early 7-0 lead.
The scrum was even-handed but the lineout was going the Blues' way. But in a rare foray into the Chichester half, the Bulls kicker showed his class to punish an unclear transgression from Chichester with a solid penalty.
The Bulls were still feeling the pressure and a loose pass was snaffled by Richard Kaye to sprint clear. Adams again completed the job with a simple conversion.
The Blues looked in control but with 15 minutes of the half left, a speculative kick down-field was being chased by a Bulls back with two Blues defenders closing in.
Second-row Ridsdale was harshly adjudged to have blocked the chaser. It was a penalty at best but the referee showed yellow.
Almost immediately, a second yellow came for handling on the floor in the kill zone and the Blues were down to 13 for ten minutes.
The Blues rose to the task as the Reds piled on the pressure. The six-man scrum held the Bulls eight and after three scrums, Tom Polhill was brought into the back row to try to stave off what looked like an inevitable score.
Remarkably, the visitors scrambled and swarmed in defence to hold off the Bulls.
Aylesford kicked a penalty as the ten minutes ran down and it was 14-6.
But as the half came to a close, a well-worked forward drive saw Aylesford go over with the conversion and Chichester now led only 14-13.
"Restart the second half strong and get back on track" was the call at half-time. But the opposite occurred when a mistake led to a missed pass on their own line, the clearing pass going to ground and a defending scrum five metres out.
The Bulls executed a textbook blindside break for the right winger to go over in the corner.
A well-struck penalty from Adams brought it back to 17-18 with 20 minutes to go.
Chichester needed to get back into the Bulls half. Adfield, Upton and Dickin came on to add energy.
Polhill and Johnson carried strongly and Adfield smashed the defenders aside as the Bulls were pushed back. A penalty was given for in at the side and Veltom tapped and ran on to power his huge frame over for a 22-18 lead.
The Bulls forced another penalty which was slotted over.
Eventually, the urgency which had been missing from Chichester all game appeared as skipper Moses Kasujja urged more effort.
The blue wall held strong with an excellent positional kick from Smallman forcing the Bulls deep into their own half.
With Chichester camped in the Bulls 22, Polhill rounded off another excellent individual performance to burrow over from short range.
Adams nailed the conversion from wide to seal the game 29-21.
Afterwards, Kasuuja said scoring so early might have led to some complacency. But the Bulls made life difficult.
Head coach Richard Isaac said: "We will take the win. It wasn't a great performance but sometimes you have to dig out the result."
Director of rugby Paul Colley said: "The Bulls will trouble teams; but we must improve."
This Saturday, Chichester host Beckenham at Oaklands Park.
CHICHESTER: Shopland, O'Hagan, Veltom, Ridsdale, Blount, Johnson, Polhill, Kasujja, Sully, Smallman, Fairbairn, Pohill T, Kaye, Seviua, Adams; Adfield, Dickin, Upton.