The in-form visitors went into the match knowing a win would virtually guarantee their future in the top-flight, while Fulham knew that anything but victory would mean little hope of survival.
The Cottagers duly sprang out of the traps with some early pressure and, on 13 minutes, striker Brian McBride should have done better than to head straight into the arms of Craig Gordon when free in the box from 12 yards out.
Seconds later Welsh midfielder Simon Davies hammered a shot just over the bar from outside the box.
Draw specialists Fulham continued to probe but had a lucky escape when referee Mark Halsey disallowed a Collins headed goal for a supposed shove on Leon Andreasen after he converted Andy Reid's dead-ball cross.
To a crescendo of noise from the Wearsider's visiting support, the tempo of the game settled down with both sides looking controlled in possession without showing any real cutting edge.
Davies again fired a shot just over the bar but it was Roy Keane's side who made the breakthrough just before half-time.
Daryl Murphy dropped a shoulder to find space under pressure from Brede Hangeland and the Sunderland man's cross was nodded past Kasey Keller by Collins, who was not to be denied on this occasion.
Within a minute of the restart Fulham carved out a clear chance, with McBride scuffing the ball across the box and Diomansy Kamara failing to convert from close in as Nyron Nosworthy helped Sunderland clear their lines.
And the visitors soon pushed upfield to extend their lead.
Jones cleverly headed a long punt from defence into the path of half-time substitute Chopra, who let the ball drop over his head and instinctively lobbed Keller with his left foot.
Desperate Fulham boss Hodgson made an instant substiution, swapping defender Andreasen with Northern Ireland striker Healy and, minutes later, replacing Paul Stalteri with Clinton Dempsey - another attacking switch.
The changes had little immediate effect and confident Sunderland pushed forward with substitute Grant Leadbitter forcing Keller into an impressive high stop from distance.
Healy did give Fulham a glimmer of hope, expertly curling the ball into the top corner past Gordon with his left foot from 20 yards.
But Sunderland hit back instantly, Murphy marauding into the box and squaring for Jones to stroke into the corner of the net from 15 yards past a stricken Fulham rearguard.
The home side battled until the final whistle and Healy guided a header just wide of the post with five minutes remaining.
But, ultimately, their lack of goals throughout the season would appear to have cost them their place in the Premier League, while Sunderland rejoice over a priceless win.
Fulham boss Roy Hodgson: "I'm pretty low at the moment. The only thing in our favour is that we are still six points behind.
"But it's a bitter blow. In the first half, just before their goal, I had every hope that our performance was going to give us a result.
"It was always uphill after their second goal. We got in good crossing positions but our final ball wasn't good enough.
"The only lifeline is that football is not played on paper - on paper it looks very, very bad for us."
Sunderland manager Roy Keane: "We can't relax. We have to make sure we have enough points so we mathematically cannot be caught.
"We didn't play particularly well. I'd say we were very bad. We didn't keep it enough but we scored at the right time.
"Chops was unlucky not to start. If you get one or two chances in the box he will score.
"David Healy scored a wonder goal and we fortunately got one quickly. On this occasion we scored at the right time. The spirit of players has been fantastic and we saw that again today."
BBC Sport Player Rater man of the match: Sunderland's Danny Collins 7.86 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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