The London side have now have now gone 16 games without a win - equalling a miserable record set 53 years ago - with a question mark hanging over the future of caretaker boss Phil Parkinson.
It all started well for the Addicks as on-loan Fulham striker Hameur Bouazza scored in spectacular fashion in the 34th minute, before Quinn grabbed a brace as United hit back after the interval.
Charlton started brightly testing Blades reserve keeper Ian Bennett - in again for Paddy Kenny, who was dropped for the Boxing Day match at Wolves for disciplinary reasons.
Bouazza fired over after neat work from Deon Burton before the pair combined again.
Bouazza cleverly chested the ball down for Burton, whose stinging drive flashed over the top.
Midfielder Lloyd Sam tried his luck with a left foot effort from 20 yards which was snuffed out by Bennett as Charlton went close again.
Charlton snatched the lead in the 34th minute when keeper Rob Elliot's long kick up field found Bouazza, who ran on to finish in spectacular fashion with a stinging left-foot strike from the edge of the area.
United should have levelled matters two minutes later when James Beattie found Webber with a deft pass, but the former Watford striker hammered the ball high into the stand from eight yards.
The Sheffield side went close again on 54 minutes when midfielder Quinn's curling free-kick finding skipper Chris Morgan at the far post, but he pulled his firm header wide.
United pulled level in the 65th minute as Webber's cross found Beattie. He laid the ball into the path of Quinn, whose low drive beat the diving Elliot.
Midfielder Grant Basey saw his free-kick cut out by the Charlton keeper before United swept ahead on 72 minutes.
Webber ran onto Quinn's free-kick before firing home at the second attempt after his first shot was blocked by Elliot.
Quinn killed of battling Charlton 12 minutes from time with an eye-catching free-kick as he squeezed his left-foot strike into the top right-hand corner.