The London rivals started the New Year's Day showdown at Craven Cottage in the relegation zone and the match was poised at 1-1 when Kevin Nolan was sent off for West Ham, putting Fulham in the driving seat.
Mohamed Diame and Sidwell swapped goals before referee Mark Clattenburg showed Nolan a red card for kicking Fernando Amorebieta two minutes before half-time.
It was the West Ham captain's second dismissal within a month after also departing early in the 4-1 defeat at Liverpool on December 7.
Fulham battered the opposition defence and were rewarded with a winner that was easily dispatched by Berbatov following an astute pass from Scott Parker.
It was just the tonic the Cottagers needed after Saturday's harrowing 6-0 defeat at Hull, a result that prompted several players to apologise for their performance.
They now sit 16th in the Premier League while West Ham remain marooned in the bottom three.
Fulham boss Rene Meulensteen reacted to the rout in East Yorkshire by making seven changes, among them the return of Berbatov and Adel Taarabt from injury.
The overhauled starting XI made a purposeful start amid awful conditions at Craven Cottage, but it was soon undone as some weak defending allowed Diame to snatch the lead.
Berbatov strayed offside and the resultant free-kick was floated long into the box where Diame was given the time and space to pick his spot and stab the ball home.
Play resumed with Fulham on the offensive but once again it was the Hammers who were the more threatening as Nolan saw a header saved by goalkeeper David Stockdale.
He did everything right by making himself big, but he knew little about the ball until it struck his right foot.
Fulham almost equalised when Modibo Maiga nodded Amorebieta's header off the line, but it was only a matter of time before the equaliser came.
Damien Duff took a corner and Sidwell, arriving late and picking a good run, powered in the header.
The awful weather further deteriorated but the rain failed to dim Fulham's endeavour as Taarabt hit the left post, ending a lively passage of play.
Craven Cottage then witnessed the unattractive side of Taarabt's game as the Moroccan playmaker performed a comical dive. Amorebieta joined in the theatrics two minutes before half-time, but this time the outcome was more damaging as Nolan was given his marching orders.
The two tangled off the ball and Nolan, taking exception to being nudged, kicked Amorebieta on the calf and the left-back tumbled. Clattenburg reached for the red card and Nolan was gone.
The second goal looked imminent as Duff steered a shot just over the crossbar and was then denied by Jussi Jaaskelainen.
Fulham continued to pepper the opposition goal and Jaaskelainen was the saviour once more as he got fingertips to an attempt by Taarabt to turn it onto the woodwork.
Taarabt had another go but there was far too much power on the follow-up.
West Ham were rocking and they finally cracked in the 66th minute when substitute Alexander Kacaniklic marauded down the left wing and found Parker.
Showing superb awareness, Parker played the ball in to the unmarked Berbatov who was on target from six yards out.
It would have finished 3-1 had substitute Darren Bent not stabbed a chance straight at Jaaskelainen when through one on one with the Finn.