The visitors took the lead on 12 minutes at Craven Cottage via Steven Caulker's header from a Peter Whittingham corner from the left.
Cardiff continued to dominate and Caulker had a goal disallowed after Aron Gunnarson was adjudged to have fouled Fulham keeper David Stockdale.
Bryan Ruiz, brought on to try and stem the Bluebirds' tide, then struck with a great left-footed effort into the top corner on the stroke of half time to bring the hosts level.
That was how the score looked to stay until substitute Mutch collected a free-kick from keeper David Marshall and slotted home from 20 yards with a sublime shot into the top corner to win it for the Bluebirds deep into injury time.
The loss piles more pressure on Fulham boss Martin Jol with his side entrenched in the drop zone.
West Brom scored a last-gasp equaliser to deny Fulham a first home win in the league since defeat of QPR on April 1 in the previous Premier League contest and Jol's men were on the receiving end of another late blow.
Fulham owner Shahid Khan has already removed the Michael Jackson statue from Craven Cottage and Jol could be the next to be toppled.
Victory over Everton in the Capital One Cup on Tuesday night helped alleviate some of the pressure on the Dutchman, who selected Darren Bent as one of five changes.
Fulham were forced into an early change when Scott Parker limped off, with Giorgos Karagounis on in his place.
Only Peter Odemwingie was retained from the Cardiff side which lost in the League Cup to West Ham and the Nigeria forward was an early nuisance to Fulham.
The blow suffered by Parker's departure was exacerbated when Fulham fell behind from a Whittingham corner, which Caulker headed in unopposed eight yards out.
It was reminiscent of the West Brom equaliser a fortnight ago which resulted in calls for Jol's departure.
The Dutchman needed a response from his team, but it was Cardiff who were lifted by the goal and Andrew Taylor had a header cleared away by Steve Sidwell.
Another Whittingham set-piece highlighted the deficiencies in the Fulham defence when Ben Turner lost Brede Hangeland to head towards goal but David Stockdale was equal to it.
Caulker had the ball in the net for a second time, but the strike was ruled out for a push on Stockdale.
Karagounis tested Marshall from the edge of the area before Jol made his second change before the interval, sending on Ruiz for Alex Kacaniklic. It appeared to be a tactical move, but it was a gamble given Ruiz has been struggling with an ankle injury of late.
But it paid off as the newcomer collected a pass infield from Sascha Riether and picked his spot with a delightful curling effort which Marshall could only watch as the ball soared into the top corner.
Ruiz celebrated by running to the bench to high-five Jol, whose team appeared less fragile in the early stages of the second half.
Yet it was Cardiff who nearly took the lead for a second time.
Mutch replaced Kim Bo-kyung and had an immediate impact, playing through Frazier Campbell who fluffed the opportunity.
Fulham then manufactured a number of chances without making the most of them.
Berbatov came close, meeting Riether's cross outside the far post but unable to find the target.
Ruiz then created a chance for Pajtim Kasami, who merely side-footed the ball tamely at Marshall, before Kasami's shot after a solo run was blocked by Caulker and the Fulham midfielder weakly hit his follow-up at Marshall.
Fulham were made to pay as Mutch brought down Marshall's long free-kick, got a lucky break off Kieran Richardson and curled the ball into the top corner to seal a massive three points for the Welsh side.