The London side, heading for League One, were on the end of as 3-1 defeat at Bramall Lane in the Championship and were humbled 4-1 by Sheffield Wednesday last week.
Webber and a Lee Hendrie free-kick, only his second for the club, ended Charlton's FA Cup campaign to leave manager Phil Parkinson to focus on the almost impossible task of avoiding the drop.
The London side, in freefall and with just one win in 20 games, almost caused an early upset, midfielder Lloyd Sam picked out Deon Burton as he pumped in a cross from the left flank, the ex-Wednesday striker's firm header was tipped over by Ian Bennett.
Darren Ambrose burst through minutes later beating centre-backs Chris Morgan and Matt Kilgallon, but the United keeper charged off his line to block the midfielder low drive with his feet.
Charlton caused more problems, Ambrose drilling in a low shot from the edge of the area which was scooped up on the line by Bennett.
United, unbeaten in five, bounced back. Billy Sharp hit a teasing cross, the ball bouncing in front of Webber who stretched but couldn't keep his close-range header on target.
Webber finally broke the deadlock in the 26th minute, the ex-Watford striker spotted Rob Elliot off his line and slammed in an angled strike from 30 yards, the ball dropping into the far corner as the keeper scrambled back.
Charlton escaped after Elliot blundered. The keeper's clearance was half blocked by Greg Halford, the ball spun to Darius Henderson, but the Charlton keeper charged across his line to smother the United striker's low shot.
United added a second in the 61st minute, substitute Hendrie, back from a loan spell at Blackpool, curling his free-kick from 25 yards around the wall and beyond the stranded Elliot.
Substitute Chris Dickson gave Charlton a lifeline in the 69th minute as he ran on to a Jonathan Fortune free-kick to beat Gary Naysmith and lob the ball over the advancing Bennett.
The Addicks then tried their best to force an equaliser, but the home side held out to secure a deserved place in round five.