The home side scored first after a Wells counter-attack ended with a pass going astray on their left after 13 minutes.
Sidcup switched the ball fluidly to the opposite wing where centre Oli Vidgeon dummied and went over from 30 metres for a converted try.
Wells responded with their own turnover, and though they were stopped just short of the line Nick Doherty reached over from the base of the ruck for an unconverted try after 24 minutes.
The visitors were on the back foot again when they conceded a penalty soon afterwards, the Wells pack collapsing two scrums.
After Frank Reynolds saw his own penalty kick slip wide, Sidcup went in at half-time with a 14-5 lead.
The second half began with a madcap 10 minutes in which four tries were scored including another controversial penalty try.
Ryan Taylor-Dennehy, replacing the injured Max Hobbs on the wing, broke from 35 metres out and found Reynolds on his inside shoulder.
He cleverly drew the full-back before giving it back to the skipper to score under the posts, then added the extras himself.
From the restart, though, Sidcup lock Freddy Ruff found a hole in the Wells ruck defence and burst clear 30 metres out before powering through a last-ditch tackle to score, Josh Twyford converting.
Back came Wells, clever play down the blind-side allowing centre Mike Doherty to power his way over for a try that was expertly converted from the left touchline by Reynolds.
After three tries in four minutes the contest was now delicately poised at 21-19, but then came the moment of controversy.
Colby McMahon kicked a stray Wells pass ahead and looked set to score before Mike Doherty, racing back, dived onto the ball a metre short of the line.
The referee, however, awarded a penalty try and showed Doherty a yellow card.
A skirmish between the two teams on the halfway line saw two more players sent to the sin-bin, Wells’ Josh Hawkins and Sidcup flanker Lennie Bush.
Twyford sent a mighty penalty kick 45 metres between the sticks to stretch the lead to 31-19 as the game entered the last ten minutes.
Then Wells were picked off while on the attack again. A looping pass to the left was intercepted by home wing Andre Patterson, who held off two defenders to run the ball in from 65 metres for a converted try.
The visitors did not look like securing the try bonus point and also lost Agy Eukaliti to the sin-bin for a no-arms tackle as time ebbed away.
Wells drop down to fifth place in the table, just two points behind Hertford but losing ground on the top three. Leaders Wimbledon lost their unbeaten record at home to second-placed Guernsey.
Wells now face Wimbledon in another tough test on Saturday [December 14, kick-off 2pm].
