Antoine Semenyo struck after just 53 seconds at Vitality Stadium on Monday, pouncing on an error from Fulham captain Antonee Robinson to claim his 10th goal of the season.
Evanilson went close to doubling the hosts' lead when he struck the woodwork before Andoni Iraola's men hung on to climb to eighth - above their opponents and Brighton on goal difference.
Ryan Sessegnon wasted the best of Fulham's chances to equalise - heading straight at Kepa Arrizabalaga in the first half - before seemingly being substituted incorrectly during the second period.
Bournemouth began the evening having taken just two points from a possible 18 during a dramatic drop-off in form since going fifth with a 3-1 success at relegated rivals Southampton on February 15.
"As soon as I got it, I knew I had space," Semenyo said of his early strike.
"I wanted to shift him one side, hopefully get on the left comfortably, and pass it in the net. I feel like I've been in that position, hit it hard, and it's gone over the bar.
"I knew when I went past him, I just wanted to pass it into the corner. As soon as he turned, I knew — get the shot away."
"The manager's been saying all week it's a make-or-break game for us," Semenyo added.
"We wanted to give it all and you can see from the celebrations it meant a lot."
Fulham keeper Bernd Leno said his side was lucky to lose by the single goal.
"It's our fault we lost today. The first 20 minutes we were not on point or at our level, we were lucky it was just 1-0," he told Sky Sports.
"(In the) second half we tried everything and were much better but Bournemouth defended well.
"We had a lot of chances, a couple of headers, dangerous crosses, but in the end the last percentage was missing."
The hosts comfortably held on during seven minutes of added time to claim an overdue victory - their first at home since thrashing Nottingham Forest 5-0 in January - and put their European push back on track.
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